Theorem syl6eqelr 2126

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

Hypotheses

Ref	Expression
syl6eqelr.1	⊢ (φ → B = A)
syl6eqelr.2	⊢ B ∈ 𝐶

Assertion

Ref	Expression
syl6eqelr	⊢ (φ → A ∈ 𝐶)

Proof of Theorem syl6eqelr

Step	Hyp	Ref	Expression
1		syl6eqelr.1	. . 3 ⊢ (φ → B = A)
2	1	eqcomd 2042	. 2 ⊢ (φ → A = B)
3		syl6eqelr.2	. 2 ⊢ B ∈ 𝐶
4	2, 3	syl6eqel 2125	1 ⊢ (φ → A ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1242   ∈ 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:  eusvnfb  4152  releldm2  5753  bren  6164  brdomg  6165  ioof  8610