Theorem eqeqan12dOLD 2760

Description: Obsolete version of eqeqan12d 2753 as of 23-Oct-2024. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
eqeqan12dOLD	⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeqan12dOLD

Step	Hyp	Ref	Expression
1		eqeqan12dOLD.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
3		eqeqan12dOLD.2	. . 3 ⊢ (𝜓 → 𝐶 = 𝐷)
4	3	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)
5	2, 4	eqeq12dOLD 2759	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2731

This theorem is referenced by: (None)