Theorem cdeqel 2981

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 2971	. . 3 |- ( x = y -> A = B )
3		cdeqeq.2	. . . 4 |- CondEq ( x = y -> C = D )
4	3	cdeqri 2971	. . 3 |- ( x = y -> C = D )
5	2, 4	eleq12d 2264	. 2 |- ( x = y -> ( A e. C <-> B e. D ) )
6	5	cdeqi 2970	1 |- CondEq ( x = y -> ( A e. C <-> B e. D ) )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2164  CondEqwcdeq 2968

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189  df-cdeq 2969

This theorem is referenced by:  nfccdeq  2983