Theorem sbccomglem 1978

Description: Lemma for sbccomg 1979.

Assertion

Ref	Expression
sbccomglem	|- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D

Proof of Theorem sbccomglem

Step	Hyp	Ref	Expression
1		sbc5g 1944	. . . 4 |- (B e. D -> ([B / y]ph <-> E.y(y = B /\ ph)))
2	1	sbcbidv 1967	. . 3 |- ((B e. D /\ A e. C) -> ([A / x][B / y]ph <-> [A / x]E.y(y = B /\ ph)))
3	2	ancoms 436	. 2 |- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [A / x]E.y(y = B /\ ph)))
4		sbc5g 1944	. . 3 |- (A e. C -> ([A / x]E.y(y = B /\ ph) <-> E.x(x = A /\ E.y(y = B /\ ph))))
5	4	adantr 389	. 2 |- ((A e. C /\ B e. D) -> ([A / x]E.y(y = B /\ ph) <-> E.x(x = A /\ E.y(y = B /\ ph))))
6		sbc5g 1944	. . . . 5 |- (A e. C -> ([A / x]ph <-> E.x(x = A /\ ph)))
7	6	sbcbidv 1967	. . . 4 |- ((A e. C /\ B e. D) -> ([B / y][A / x]ph <-> [B / y]E.x(x = A /\ ph)))
8		sbc5g 1944	. . . . 5 |- (B e. D -> ([B / y]E.x(x = A /\ ph) <-> E.y(y = B /\ E.x(x = A /\ ph))))
9	8	adantl 388	. . . 4 |- ((A e. C /\ B e. D) -> ([B / y]E.x(x = A /\ ph) <-> E.y(y = B /\ E.x(x = A /\ ph))))
10	7, 9	bitr2d 527	. . 3 |- ((A e. C /\ B e. D) -> (E.y(y = B /\ E.x(x = A /\ ph)) <-> [B / y][A / x]ph))
11		excom 1042	. . . 4 |- (E.xE.y(x = A /\ (y = B /\ ph)) <-> E.yE.x(x = A /\ (y = B /\ ph)))
12		exdistr 1304	. . . 4 |- (E.xE.y(x = A /\ (y = B /\ ph)) <-> E.x(x = A /\ E.y(y = B /\ ph)))
13		an12 483	. . . . . . 7 |- ((x = A /\ (y = B /\ ph)) <-> (y = B /\ (x = A /\ ph)))
14	13	exbii 1047	. . . . . 6 |- (E.x(x = A /\ (y = B /\ ph)) <-> E.x(y = B /\ (x = A /\ ph)))
15		19.42v 1303	. . . . . 6 |- (E.x(y = B /\ (x = A /\ ph)) <-> (y = B /\ E.x(x = A /\ ph)))
16	14, 15	bitr 173	. . . . 5 |- (E.x(x = A /\ (y = B /\ ph)) <-> (y = B /\ E.x(x = A /\ ph)))
17	16	exbii 1047	. . . 4 |- (E.yE.x(x = A /\ (y = B /\ ph)) <-> E.y(y = B /\ E.x(x = A /\ ph)))
18	11, 12, 17	3bitr3 181	. . 3 |- (E.x(x = A /\ E.y(y = B /\ ph)) <-> E.y(y = B /\ E.x(x = A /\ ph)))
19	10, 18	syl5bb 530	. 2 |- ((A e. C /\ B e. D) -> (E.x(x = A /\ E.y(y = B /\ ph)) <-> [B / y][A / x]ph))
20	3, 5, 19	3bitrd 542	1 |- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  [wsbc 1166

This theorem is referenced by:  sbccomg 1979

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-or 224  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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