Theorem pm5.33 573

Description: Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm5.33	⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))

Proof of Theorem pm5.33

Step	Hyp	Ref	Expression
1		ibar 295	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
2	1	imbi1d 229	. 2 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ ((𝜑 ∧ 𝜓) → 𝜒)))
3	2	pm5.32i 441	1 ⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)