Theorem pm5.4 249

Description: Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
pm5.4	⊢ ((𝜑 → (𝜑 → 𝜓)) ↔ (𝜑 → 𝜓))

Proof of Theorem pm5.4

Step	Hyp	Ref	Expression
1		pm2.43 53	. 2 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
2		ax-1 6	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜓)))
3	1, 2	impbii 126	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  imbibi  252  sbequ8  1861  moabs  2094  isprm4  12287  limcdifap  14898