Theorem frege52aid 45001

Description: The case when the content of 𝜑 is identical with the content of 𝜓 and in which 𝜑 is affirmed and 𝜓 is denied does not take place. Identical to biimp 218. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege52aid	⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))

Proof of Theorem frege52aid

Step	Hyp	Ref	Expression
1		ax-frege52a 45000	. 2 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, ⊤, ⊥) → if-(𝜓, ⊤, ⊥)))
2		ifpid2 44615	. 2 ⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥))
3		ifpid2 44615	. 2 ⊢ (𝜓 ↔ if-(𝜓, ⊤, ⊥))
4	1, 2, 3	3imtr4g 299	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  if-wif 1078  ⊤wtru 1570  ⊥wfal 1581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege52a 45000

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-tru 1572  df-fal 1582

This theorem is used by:  frege53aid  45002  frege57aid  45015  frege75  45081  frege89  45095  frege105  45111