Theorem syl5eqelr 2125

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 2044	. 2 ⊢ 𝐴 = 𝐵
3		syl5eqelr.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
4	2, 3	syl5eqel 2124	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  dmrnssfld  4595  cnvexg  4855  opabbrex  5549  offval  5719  resfunexgALT  5737  abrexexg  5745  abrexex2g  5747  opabex3d  5748  nqprlu  6645  iccshftr  8862  iccshftl  8864  iccdil  8866  icccntr  8868