Theorem eqeq12dOLD 2757

Description: Obsolete version of eqeq12d 2753 as of 23-Oct-2024. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
eqeq12dOLD.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqeq12dOLD.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
eqeq12dOLD	⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeq12dOLD

Step	Hyp	Ref	Expression
1		eqeq12dOLD.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eqeq12dOLD.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		eqeq12OLD 2756	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	syl2anc 587	1 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2729

This theorem is referenced by:  eqeqan12dOLD  2758