Theorem pm5.42 553

Description: Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm5.42	⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))

Proof of Theorem pm5.42

Step	Hyp	Ref	Expression
1		ibar 538	. . 3 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒)))
2	1	imbi2d 343	. 2 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → (𝜑 ∧ 𝜒))))
3	2	pm5.74i 274	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

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

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  anc2l  563  imdistan  578