Theorem eleq12d 2421

Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Hypotheses

Ref	Expression
eleq1d.1	⊢ (φ → A = B)
eleq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
eleq12d	⊢ (φ → (A ∈ C ↔ B ∈ D))

Proof of Theorem eleq12d

Step	Hyp	Ref	Expression
1		eleq12d.2	. . 3 ⊢ (φ → C = D)
2	1	eleq2d 2420	. 2 ⊢ (φ → (A ∈ C ↔ A ∈ D))
3		eleq1d.1	. . 3 ⊢ (φ → A = B)
4	3	eleq1d 2419	. 2 ⊢ (φ → (A ∈ D ↔ B ∈ D))
5	2, 4	bitrd 244	1 ⊢ (φ → (A ∈ C ↔ B ∈ D))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710

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

This theorem is used by:  cbvraldva2  2840  cbvrexdva2  2841  ru  3046  sbcel12g  3152  cbvralcsf  3199  cbvreucsf  3201  cbvrabcsf  3202  nenpw1pwlem2  6086  nmembers1  6272