Theorem sbccomglem 2525

Description: Lemma for sbccomg 2526.

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 2470	. . . 4 |- (B e. D -> ([B / y]ph <-> E.y(y = B /\ ph)))
2	1	sbcbidv 2505	. . 3 |- ((B e. D /\ A e. C) -> ([A / x][B / y]ph <-> [A / x]E.y(y = B /\ ph)))
3	2	ancoms 484	. 2 |- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [A / x]E.y(y = B /\ ph)))
4		sbc5g 2470	. . 3 |- (A e. C -> ([A / x]E.y(y = B /\ ph) <-> E.x(x = A /\ E.y(y = B /\ ph))))
5	4	adantr 425	. 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 2470	. . . . 5 |- (A e. C -> ([A / x]ph <-> E.x(x = A /\ ph)))
7	6	sbcbidv 2505	. . . 4 |- ((A e. C /\ B e. D) -> ([B / y][A / x]ph <-> [B / y]E.x(x = A /\ ph)))
8		sbc5g 2470	. . . . 5 |- (B e. D -> ([B / y]E.x(x = A /\ ph) <-> E.y(y = B /\ E.x(x = A /\ ph))))
9	8	adantl 424	. . . 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 588	. . 3 |- ((A e. C /\ B e. D) -> (E.y(y = B /\ E.x(x = A /\ ph)) <-> [B / y][A / x]ph))
11		excom 1393	. . . 4 |- (E.xE.y(x = A /\ (y = B /\ ph)) <-> E.yE.x(x = A /\ (y = B /\ ph)))
12		exdistr 1689	. . . 4 |- (E.xE.y(x = A /\ (y = B /\ ph)) <-> E.x(x = A /\ E.y(y = B /\ ph)))
13		an12 542	. . . . . . 7 |- ((x = A /\ (y = B /\ ph)) <-> (y = B /\ (x = A /\ ph)))
14	13	exbii 1398	. . . . . 6 |- (E.x(x = A /\ (y = B /\ ph)) <-> E.x(y = B /\ (x = A /\ ph)))
15		19.42v 1688	. . . . . 6 |- (E.x(y = B /\ (x = A /\ ph)) <-> (y = B /\ E.x(x = A /\ ph)))
16	14, 15	bitri 190	. . . . 5 |- (E.x(x = A /\ (y = B /\ ph)) <-> (y = B /\ E.x(x = A /\ ph)))
17	16	exbii 1398	. . . 4 |- (E.yE.x(x = A /\ (y = B /\ ph)) <-> E.y(y = B /\ E.x(x = A /\ ph)))
18	11, 12, 17	3bitr3i 198	. . 3 |- (E.x(x = A /\ E.y(y = B /\ ph)) <-> E.y(y = B /\ E.x(x = A /\ ph)))
19	10, 18	syl5bb 591	. 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 603	1 |- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534

This theorem is referenced by:  sbccomg 2526

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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