Theorem syl6eleq 2443

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

Hypotheses

Ref	Expression
syl6eleq.1	⊢ (φ → A ∈ B)
syl6eleq.2	⊢ B = C

Assertion

Ref	Expression
syl6eleq	⊢ (φ → A ∈ C)

Proof of Theorem syl6eleq

Step	Hyp	Ref	Expression
1		syl6eleq.1	. 2 ⊢ (φ → A ∈ B)
2		syl6eleq.2	. . 3 ⊢ B = C
3	2	a1i 10	. 2 ⊢ (φ → B = C)
4	1, 3	eleqtrd 2429	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:  syl6eleqr  2444  prid2g  3827  sfinltfin  4536  vinf  4556  ndmfvrcl  5351  enmap2lem3  6066  enmap1lem3  6072  enprmaplem3  6079  enprmaplem5  6081  enprmaplem6  6082  eqncg  6127  ncseqnc  6129