Theorem hbsbcgd 1974

Description: Deduction version of hbsbcg 1941.

Hypotheses

Ref	Expression
hbsbcgd.1	|- (ph -> A.xph)
hbsbcgd.2	|- (ph -> A.yph)
hbsbcgd.3	|- (ph -> (z e. A -> A.x z e. A))
hbsbcgd.4	|- (ph -> (ps -> A.xps))

Assertion

Ref	Expression
hbsbcgd	|- ((ph /\ A e. C) -> ([A / y]ps -> A.x[A / y]ps))

Distinct variable groups:   z,A   ph,z   x,z

Proof of Theorem hbsbcgd

Step	Hyp	Ref	Expression
1		ax-4 970	. . . . . . . . 9 |- (A.x z e. A -> z e. A)
2		hbsbcgd.3	. . . . . . . . 9 |- (ph -> (z e. A -> A.x z e. A))
3	1, 2	impbid2 516	. . . . . . . 8 |- (ph -> (A.x z e. A <-> z e. A))
4	3	abbidv 1569	. . . . . . 7 |- (ph -> {z | A.x z e. A} = {z | z e. A})
5		eleq1 1526	. . . . . . . . 9 |- (z = w -> (z e. A <-> w e. A))
6	5	albidv 1273	. . . . . . . 8 |- (z = w -> (A.x z e. A <-> A.x w e. A))
7	6	cbvabv 1900	. . . . . . 7 |- {z | A.x z e. A} = {w | A.x w e. A}
8		abid2 1572	. . . . . . 7 |- {z | z e. A} = A
9	4, 7, 8	3eqtr3g 1522	. . . . . 6 |- (ph -> {w | A.x w e. A} = A)
10	9	eleq1d 1532	. . . . 5 |- (ph -> ({w | A.x w e. A} e. V <-> A e. V))
11	10	biimpar 417	. . . 4 |- ((ph /\ A e. V) -> {w | A.x w e. A} e. V)
12		hba1 1000	. . . . . 6 |- (A.x w e. A -> A.xA.x w e. A)
13	12	hbab 1460	. . . . 5 |- (z e. {w | A.x w e. A} -> A.x z e. {w | A.x w e. A})
14		hba1 1000	. . . . 5 |- (A.xps -> A.xA.xps)
15	13, 14	hbsbcg 1941	. . . 4 |- ({w | A.x w e. A} e. V -> ([{w | A.x w e. A} / y]A.xps -> A.x[{w | A.x w e. A} / y]A.xps))
16	11, 15	syl 10	. . 3 |- ((ph /\ A e. V) -> ([{w | A.x w e. A} / y]A.xps -> A.x[{w | A.x w e. A} / y]A.xps))
17	2	19.21aiv 1281	. . . . . 6 |- (ph -> A.z(z e. A -> A.x z e. A))
18		abidhb 1903	. . . . . 6 |- (A.z(z e. A -> A.x z e. A) -> {w | A.x w e. A} = A)
19		dfsbcq 1933	. . . . . 6 |- ({w | A.x w e. A} = A -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
20	17, 18, 19	3syl 20	. . . . 5 |- (ph -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
21	20	adantr 389	. . . 4 |- ((ph /\ A e. V) -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
22		hbsbcgd.2	. . . . 5 |- (ph -> A.yph)
23		ax-4 970	. . . . . 6 |- (A.xps -> ps)
24		hbsbcgd.4	. . . . . 6 |- (ph -> (ps -> A.xps))
25	23, 24	impbid2 516	. . . . 5 |- (ph -> (A.xps <-> ps))
26	22, 25	sbcbid 1966	. . . 4 |- ((ph /\ A e. V) -> ([A / y]A.xps <-> [A / y]ps))
27	21, 26	bitrd 526	. . 3 |- ((ph /\ A e. V) -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]ps))
28		hbsbcgd.1	. . . . . . 7 |- (ph -> A.xph)
29	28	a1d 12	. . . . . 6 |- (ph -> (ph -> A.xph))
30		ax-17 968	. . . . . . . 8 |- (z e. V -> A.x z e. V)
31	30	a1i 8	. . . . . . 7 |- (ph -> (z e. V -> A.x z e. V))
32	28, 2, 31	hbeld 1905	. . . . . 6 |- (ph -> (A e. V -> A.x A e. V))
33	29, 32	hband 1107	. . . . 5 |- (ph -> ((ph /\ A e. V) -> A.x(ph /\ A e. V)))
34	33	anabsi5 494	. . . 4 |- ((ph /\ A e. V) -> A.x(ph /\ A e. V))
35	34, 27	albid 1100	. . 3 |- ((ph /\ A e. V) -> (A.x[{w | A.x w e. A} / y]A.xps <-> A.x[A / y]ps))
36	16, 27, 35	3imtr3d 540	. 2 |- ((ph /\ A e. V) -> ([A / y]ps -> A.x[A / y]ps))
37		elisset 1808	. 2 |- (A e. C -> A e. V)
38	36, 37	sylan2 451	1 |- ((ph /\ A e. C) -> ([A / y]ps -> A.x[A / y]ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166  {cab 1456  Vcvv 1802

This theorem is referenced by:  hbcsbgd 2018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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