Theorem pm5.32d 445

Description: Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)

Hypothesis

Ref	Expression
pm5.32d.1	⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))

Assertion

Ref	Expression
pm5.32d	⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))

Proof of Theorem pm5.32d

Step	Hyp	Ref	Expression
1		pm5.32d.1	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
2		bi1 117	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃))
3	1, 2	syl6 33	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4	3	imdistand 443	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
5		bi2 129	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
6	1, 5	syl6 33	. . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
7	6	imdistand 443	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜓 ∧ 𝜒)))
8	4, 7	impbid 128	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.32rd  446  pm5.32da  447  pm5.32  448  anbi2d  459  cbvex2  1894  cores  5042  isoini  5719  mpoeq123  5830  genpassl  7332  genpassu  7333  fzind  9166  btwnz  9170  elfzm11  9871  isprm2  11798  isprm3  11799