Theorem syl5eleqr 2127

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

Hypotheses

Ref	Expression
syl5eleqr.1	⊢ 𝐴 ∈ 𝐵
syl5eleqr.2	⊢ (𝜑 → 𝐶 = 𝐵)

Assertion

Ref	Expression
syl5eleqr	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem syl5eleqr

Step	Hyp	Ref	Expression
1		syl5eleqr.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		syl5eleqr.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2045	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	syl5eleq 2126	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:  rabsnt  3445  0elnn  4340  tfrexlem  5948  rdgtfr  5961  rdgruledefgg  5962