Theorem frege55lem1a 44620

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem1a	⊢ ((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓 ↔ 𝜑)))

Proof of Theorem frege55lem1a

Step	Hyp	Ref	Expression
1		frege54cor0a 44617	. . 3 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
2	1	biimpri 231	. 2 ⊢ (if-(𝜓, 𝜑, ¬ 𝜑) → (𝜓 ↔ 𝜑))
3	2	imim2i 17	1 ⊢ ((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓 ↔ 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 44584

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079

This theorem is used by:  frege55cor1a  44623