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Theorem syl6eqelr 2442

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 2358	. 2 ⊢ (φ → A = B)
3		syl6eqelr.2	. 2 ⊢ B ∈ C
4	2, 3	syl6eqel 2441	1 ⊢ (φ → A ∈ C)

Colors of variables: wff setvar class

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

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: cnvkexg 4286 p6exg 4290 ssetkex 4294 sikexg 4296 ins2kexg 4305 ins3kexg 4306 0cnelphi 4597 mapprc 6004 enmap1lem5 6073 nenpw1pwlem2 6085 sbthlem3 6205