Theorem eleq12i 1539

Description: Inference from equality to equivalence of membership.

Hypotheses

Ref	Expression
eleq1i.1	|- A = B
eleq12i.2	|- C = D

Assertion

Ref	Expression
eleq12i	|- (A e. C <-> B e. D)

Proof of Theorem eleq12i

Step	Hyp	Ref	Expression
1		eleq12i.2	. . 3 |- C = D
2	1	eleq2i 1538	. 2 |- (A e. C <-> A e. D)
3		eleq1i.1	. . 3 |- A = B
4	3	eleq1i 1537	. 2 |- (A e. D <-> B e. D)
5	2, 4	bitr 173	1 |- (A e. C <-> B e. D)

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958

This theorem is referenced by:  sbcel12g 2011  1q 5057  0r 5189  1r 5190  m1r 5191  fsumshft 7031  ispgrag 10779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472

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