Theorem syl5eleqr 2204

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 2120	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	syl5eleq 2203	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1314   ∈ wcel 1463

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111

This theorem is referenced by:  rabsnt  3564  0elnn  4492  tfrexlem  6185  rdgtfr  6225  rdgruledefgg  6226  hashinfom  10417  ennnfonelemhom  11773  exmid1stab  12887