Theorem eqeqan1dOLD 2844

Description: Implication of introducing a new equality. Obsolete as of 14-Feb-2023. Use eqeqan12d 2841 instead. (Contributed by Peter Mazsa, 17-Apr-2019.) (Proof shortened by AV, 10-Feb-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
eqeqan1dOLD.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eqeqan1dOLD	⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeqan1dOLD

Step	Hyp	Ref	Expression
1		eqeqan1dOLD.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		id 22	. 2 ⊢ (𝐶 = 𝐷 → 𝐶 = 𝐷)
3	1, 2	eqeqan12d 2841	1 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-cleq 2818

This theorem is referenced by: (None)