Theorem syl5eqelr 2438

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 2357	. 2 ⊢ A = B
3		syl5eqelr.2	. 2 ⊢ (φ → B ∈ C)
4	2, 3	syl5eqel 2437	1 ⊢ (φ → A ∈ C)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  xpkexg  4289  pw1exb  4327  nncaddccl  4420  0cminle  4462  vfinspsslem1  4551  opexb  4604  cnvexb  5104  epprc  5828  frds  5936  ovmuc  6131  ovcelem1  6172  ce2t  6236  addccan2nclem2  6265  fnfreclem1  6318  elscan  6331