Theorem eqeqan12dALT 2761

Description: Alternate proof of eqeqan12d 2753. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eqeqan12dOLD.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqeqan12dOLD.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
eqeqan12dALT	⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeqan12dALT

Step	Hyp	Ref	Expression
1		eqeqan12dOLD.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eqeqan12dOLD.2	. 2 ⊢ (𝜓 → 𝐶 = 𝐷)
3		eqeq12 2756	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	syl2an 595	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-cleq 2731

This theorem is referenced by: (None)