Theorem brcogw 5290

Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
brcogw	|- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Proof of Theorem brcogw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 1058	. 2 |- ( ( A e. V /\ B e. W /\ X e. Z ) -> ( A e. V /\ B e. W ) )
2		breq2 4657	. . . . . 6 |- ( x = X -> ( A D x <-> A D X ) )
3		breq1 4656	. . . . . 6 |- ( x = X -> ( x C B <-> X C B ) )
4	2, 3	anbi12d 747	. . . . 5 |- ( x = X -> ( ( A D x /\ x C B ) <-> ( A D X /\ X C B ) ) )
5	4	spcegv 3294	. . . 4 |- ( X e. Z -> ( ( A D X /\ X C B ) -> E. x ( A D x /\ x C B ) ) )
6	5	imp 445	. . 3 |- ( ( X e. Z /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
7	6	3ad2antl3 1225	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
8		brcog 5288	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
9	8	biimpar 502	. 2 |- ( ( ( A e. V /\ B e. W ) /\ E. x ( A D x /\ x C B ) ) -> A ( C o. D ) B )
10	1, 7, 9	syl2an2r 876	1 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   class class class wbr 4653   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  utop2nei  22054  utop3cls  22055  iunrelexpuztr  38011  frege96d  38041  frege98d  38045