Theorem eqeqan2d 38738

Description: Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)

Hypothesis

Ref	Expression
eqeqan2d.1	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
eqeqan2d	⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeqan2d

Step	Hyp	Ref	Expression
1		eqeqan2d.1	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
2		eqeq12 2779	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
3	1, 2	sylan2 602	1 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-cleq 2754

This theorem is referenced by:  ecqmap  38945