Theorem hbsbc1gd 1983

Description: Deduction version of hbsbc1g 1948.

Hypotheses

Ref	Expression
hbsbc1gd.1	|- (ph -> A.xph)
hbsbc1gd.2	|- (ph -> (y e. A -> A.x y e. A))

Assertion

Ref	Expression
hbsbc1gd	|- ((ph /\ A e. B) -> ([A / x]ps -> A.x[A / x]ps))

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

Proof of Theorem hbsbc1gd

Step	Hyp	Ref	Expression
1		ax-4 973	. . . . . . . . 9 |- (A.x y e. A -> y e. A)
2		hbsbc1gd.2	. . . . . . . . 9 |- (ph -> (y e. A -> A.x y e. A))
3	1, 2	impbid2 518	. . . . . . . 8 |- (ph -> (A.x y e. A <-> y e. A))
4	3	abbidv 1577	. . . . . . 7 |- (ph -> {y | A.x y e. A} = {y | y e. A})
5		eleq1 1534	. . . . . . . . 9 |- (y = z -> (y e. A <-> z e. A))
6	5	albidv 1278	. . . . . . . 8 |- (y = z -> (A.x y e. A <-> A.x z e. A))
7	6	cbvabv 1909	. . . . . . 7 |- {y | A.x y e. A} = {z | A.x z e. A}
8		abid2 1580	. . . . . . 7 |- {y | y e. A} = A
9	4, 7, 8	3eqtr3g 1530	. . . . . 6 |- (ph -> {z | A.x z e. A} = A)
10	9	eleq1d 1540	. . . . 5 |- (ph -> ({z | A.x z e. A} e. V <-> A e. V))
11	10	biimpar 417	. . . 4 |- ((ph /\ A e. V) -> {z | A.x z e. A} e. V)
12		hba1 1003	. . . . . 6 |- (A.x z e. A -> A.xA.x z e. A)
13	12	hbab 1467	. . . . 5 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
14	13	hbsbc1g 1948	. . . 4 |- ({z | A.x z e. A} e. V -> ([{z | A.x z e. A} / x]ps -> A.x[{z | A.x z e. A} / x]ps))
15	11, 14	syl 10	. . 3 |- ((ph /\ A e. V) -> ([{z | A.x z e. A} / x]ps -> A.x[{z | A.x z e. A} / x]ps))
16	2	19.21aiv 1286	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
17		abidhb 1912	. . . . 5 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
18		dfsbcq 1943	. . . . 5 |- ({z | A.x z e. A} = A -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
19	16, 17, 18	3syl 20	. . . 4 |- (ph -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
20	19	adantr 389	. . 3 |- ((ph /\ A e. V) -> ([{z | A.x z e. A} / x]ps <-> [A / x]ps))
21		hbsbc1gd.1	. . . . . . 7 |- (ph -> A.xph)
22	21	a1d 12	. . . . . 6 |- (ph -> (ph -> A.xph))
23		ax-17 971	. . . . . . . 8 |- (y e. V -> A.x y e. V)
24	23	a1i 8	. . . . . . 7 |- (ph -> (y e. V -> A.x y e. V))
25	21, 2, 24	hbeld 1914	. . . . . 6 |- (ph -> (A e. V -> A.x A e. V))
26	22, 25	hband 1111	. . . . 5 |- (ph -> ((ph /\ A e. V) -> A.x(ph /\ A e. V)))
27	26	anabsi5 495	. . . 4 |- ((ph /\ A e. V) -> A.x(ph /\ A e. V))
28	27, 20	albid 1104	. . 3 |- ((ph /\ A e. V) -> (A.x[{z | A.x z e. A} / x]ps <-> A.x[A / x]ps))
29	15, 20, 28	3imtr3d 542	. 2 |- ((ph /\ A e. V) -> ([A / x]ps -> A.x[A / x]ps))
30		elisset 1817	. 2 |- (A e. B -> A e. V)
31	29, 30	sylan2 451	1 |- ((ph /\ A e. B) -> ([A / x]ps -> A.x[A / x]ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811

This theorem is referenced by:  hbcsb1gd 2027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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