Theorem pm5.32d 450

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		biimp 118	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃))
3	1, 2	syl6 33	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4	3	imdistand 447	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
5		biimpr 130	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
6	1, 5	syl6 33	. . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
7	6	imdistand 447	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜓 ∧ 𝜒)))
8	4, 7	impbid 129	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.32rd  451  pm5.32da  452  pm5.32  453  anbi2d  464  cbvex2  1947  cores  5192  isoini  5897  mpoeq123  6014  genpassl  7650  genpassu  7651  fzind  9501  btwnz  9505  elfzm11  10226  isprm2  12489  isprm3  12490  modprminv  12622  modprminveq  12623