Theorem rspceov 5556

Description: A frequently used special case of rspc2ev 2963 for operation values. (Contributed by set.mm contributors, 21-Mar-2007.)

Assertion

Ref	Expression
rspceov	|- ((C e. A /\ D e. B /\ S = (CFD)) -> E.x e. A E.y e. B S = (xFy))

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

Allowed substitution hints:   A(y)   D(x)

Proof of Theorem rspceov

Step	Hyp	Ref	Expression
1		oveq1 5530	. . 3 |- (x = C -> (xFy) = (CFy))
2	1	eqeq2d 2364	. 2 |- (x = C -> (S = (xFy) <-> S = (CFy)))
3		oveq2 5531	. . 3 |- (y = D -> (CFy) = (CFD))
4	3	eqeq2d 2364	. 2 |- (y = D -> (S = (CFy) <-> S = (CFD)))
5	2, 4	rspc2ev 2963	1 |- ((C e. A /\ D e. B /\ S = (CFD)) -> E.x e. A E.y e. B S = (xFy))

Syntax hints:   -> wi 4   /\ w3a 934   = wceq 1642   e. wcel 1710  E.wrex 2615  (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: (None)