Theorem eleq12d 2105

Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Hypotheses

Ref	Expression
eleq1d.1	|- (ph -> A = B)
eleq12d.2	|- (ph -> C = D)

Assertion

Ref	Expression
eleq12d	|- (ph -> (A e. C <-> B e. D))

Proof of Theorem eleq12d

Step	Hyp	Ref	Expression
1		eleq12d.2	. . 3 |- (ph -> C = D)
2	1	eleq2d 2104	. 2 |- (ph -> (A e. C <-> A e. D))
3		eleq1d.1	. . 3 |- (ph -> A = B)
4	3	eleq1d 2103	. 2 |- (ph -> (A e. D <-> B e. D))
5	2, 4	bitrd 177	1 |- (ph -> (A e. C <-> B e. D))

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1242   e. wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  cbvraldva2  2531  cbvrexdva2  2532  cdeqel  2754  ru  2757  sbcel12g  2859  cbvralcsf  2902  cbvrexcsf  2903  cbvreucsf  2904  cbvrabcsf  2905  elvvuni  4347  elrnmpt1  4528  smoeq  5846  smores  5848  smores2  5850  iordsmo  5853  nnaordi  6017  nnaordr  6019  ltapig  6322  ltmpig  6323  fzsubel  8693  elfzp1b  8729