Theorem syl5eqelr 2122

Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl5eqelr.1	|- B = A
syl5eqelr.2	|- (ph -> B e. C)

Assertion

Ref	Expression
syl5eqelr	|- (ph -> A e. C)

Proof of Theorem syl5eqelr

Step	Hyp	Ref	Expression
1		syl5eqelr.1	. . 3 |- B = A
2	1	eqcomi 2041	. 2 |- A = B
3		syl5eqelr.2	. 2 |- (ph -> B e. C)
4	2, 3	syl5eqel 2121	1 |- (ph -> A e. C)

Syntax hints:   -> wi 4   = 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:  dmrnssfld  4538  cnvexg  4798  opabbrex  5491  offval  5661  resfunexgALT  5679  abrexexg  5687  abrexex2g  5689  opabex3d  5690  nqprlu  6530  iccshftr  8632  iccshftl  8634  iccdil  8636  icccntr  8638