Theorem caovcld 5622

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

Hypothesis

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

Assertion

Ref	Expression
caovcld	|- ((ph /\ (A e. C /\ B e. D)) -> (AFB) 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 caovcld

Step	Hyp	Ref	Expression
1		caovcld.1	. . 3 |- ((ph /\ (x e. C /\ y e. D)) -> (xFy) e. E)
2	1	ralrimivva 2706	. 2 |- (ph -> A.x e. C A.y e. D (xFy) e. E)
3		oveq1 5530	. . . 4 |- (x = A -> (xFy) = (AFy))
4	3	eleq1d 2419	. . 3 |- (x = A -> ((xFy) e. E <-> (AFy) e. E))
5		oveq2 5531	. . . 4 |- (y = B -> (AFy) = (AFB))
6	5	eleq1d 2419	. . 3 |- (y = B -> ((AFy) e. E <-> (AFB) e. E))
7	4, 6	rspc2v 2961	. 2 |- ((A e. C /\ B e. D) -> (A.x e. C A.y e. D (xFy) e. E -> (AFB) e. E))
8	2, 7	mpan9 455	1 |- ((ph /\ (A e. C /\ B e. D)) -> (AFB) e. E)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614  (class class class)co 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526

This theorem is referenced by:  caovcl  5623