Theorem syl5eqelr 2166

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

Hypotheses

Ref	Expression
syl5eqelr.1	⊢ 𝐵 = 𝐴
syl5eqelr.2	⊢ (𝜑 → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
syl5eqelr	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem syl5eqelr

Step	Hyp	Ref	Expression
1		syl5eqelr.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	eqcomi 2085	. 2 ⊢ 𝐴 = 𝐵
3		syl5eqelr.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	syl5eqel 2165	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  dmrnssfld  4613  cnvexg  4875  opabbrex  5569  offval  5739  resfunexgALT  5757  abrexexg  5765  abrexex2g  5767  opabex3d  5768  nqprlu  6737  iccshftr  9016  iccshftl  9018  iccdil  9020  icccntr  9022  exprmfct  10519