Theorem caovclg 6826

Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)

Hypothesis

Ref	Expression
caovclg.1	|- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )

Assertion

Ref	Expression
caovclg	|- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Distinct variable groups:   x, y, A   y, B   x, C, y   x, D, y   x, E, y   ph, x, y   x, F, y

Allowed substitution hint:   B( x)

Proof of Theorem caovclg

Step	Hyp	Ref	Expression
1		caovclg.1	. . 3 |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )
2	1	ralrimivva 2971	. 2 |- ( ph -> A. x e. C A. y e. D ( x F y ) e. E )
3		oveq1 6657	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	eleq1d 2686	. . 3 |- ( x = A -> ( ( x F y ) e. E <-> ( A F y ) e. E ) )
5		oveq2 6658	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eleq1d 2686	. . 3 |- ( y = B -> ( ( A F y ) e. E <-> ( A F B ) e. E ) )
7	4, 6	rspc2v 3322	. 2 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ( x F y ) e. E -> ( A F B ) e. E ) )
8	2, 7	mpan9 486	1 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912  (class class class)co 6650

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  caovcld  6827  caovcl  6828  grprinvd  6873  seqcl2  12819  seqcaopr  12838  ercpbl  16209  gsumpropd2lem  17273  imasmnd2  17327  imasgrp2  17530  gsumzaddlem  18321  imasring  18619  qusrhm  19237  mplind  19502  plymullem  23972