Theorem elabreximdv 6330

Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Hypotheses

Ref	Expression
elabreximdv.1	|- ( A = B -> ( ch <-> ps ) )
elabreximdv.2	|- ( ph -> A e. V )
elabreximdv.3	|- ( ( ph /\ x e. C ) -> ps )

Assertion

Ref	Expression
elabreximdv	|- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch )

Distinct variable groups:   x, y, A   y, B   x, C, y   ch, x   ph, x

Allowed substitution hints:   ph( y)   ps( x, y)   ch( y)   B( x)   V( x, y)

Proof of Theorem elabreximdv

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 |- F/ x ph
2		nfv 1577	. 2 |- F/ x ch
3		elabreximdv.1	. 2 |- ( A = B -> ( ch <-> ps ) )
4		elabreximdv.2	. 2 |- ( ph -> A e. V )
5		elabreximdv.3	. 2 |- ( ( ph /\ x e. C ) -> ps )
6	1, 2, 3, 4, 5	elabreximd 6329	1 |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220   E.wrex 2523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817

This theorem is referenced by: (None)