Theorem eleq2s 2445

Description: Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
eleq2s.1	⊢ (A ∈ B → φ)
eleq2s.2	⊢ C = B

Assertion

Ref	Expression
eleq2s	⊢ (A ∈ C → φ)

Proof of Theorem eleq2s

Step	Hyp	Ref	Expression
1		eleq2s.2	. . 3 ⊢ C = B
2	1	eleq2i 2417	. 2 ⊢ (A ∈ C ↔ A ∈ B)
3		eleq2s.1	. 2 ⊢ (A ∈ B → φ)
4	2, 3	sylbi 187	1 ⊢ (A ∈ C → φ)

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:  elxpi  4801  optocl  4839  ecexr  5951  ectocld  5992  ecoptocl  5997  nulnnc  6119  ncprc  6125  elnc  6126