Theorem elabreximdv 29349

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 1843	. 2 |- F/ x ph
2		nfv 1843	. 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 29348	1 |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913

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-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-v 3202

This theorem is referenced by: (None)