Theorem cdeqel 3431

Description: Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
cdeqeq.1	|- CondEq ( x = y -> A = B )
cdeqeq.2	|- CondEq ( x = y -> C = D )

Assertion

Ref	Expression
cdeqel	|- CondEq ( x = y -> ( A e. C <-> B e. D ) )

Proof of Theorem cdeqel

Step	Hyp	Ref	Expression
1		cdeqeq.1	. . . 4 |- CondEq ( x = y -> A = B )
2	1	cdeqri 3421	. . 3 |- ( x = y -> A = B )
3		cdeqeq.2	. . . 4 |- CondEq ( x = y -> C = D )
4	3	cdeqri 3421	. . 3 |- ( x = y -> C = D )
5	2, 4	eleq12d 2695	. 2 |- ( x = y -> ( A e. C <-> B e. D ) )
6	5	cdeqi 3420	1 |- CondEq ( x = y -> ( A e. C <-> B e. D ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990  CondEqwcdeq 3418

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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-cdeq 3419

This theorem is referenced by:  nfccdeq  3433