Theorem pm5.32d 423

Description: Distribution of implication over biconditional (deduction rule). (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 111	. . . 4 ⊢ ((χ ↔ θ) → (χ → θ))
3	1, 2	syl6 29	. . 3 ⊢ (φ → (ψ → (χ → θ)))
4	3	imdistand 421	. 2 ⊢ (φ → ((ψ ∧ χ) → (ψ ∧ θ)))
5		bi2 121	. . . 4 ⊢ ((χ ↔ θ) → (θ → χ))
6	1, 5	syl6 29	. . 3 ⊢ (φ → (ψ → (θ → χ)))
7	6	imdistand 421	. 2 ⊢ (φ → ((ψ ∧ θ) → (ψ ∧ χ)))
8	4, 7	impbid 120	1 ⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm5.32rd  424  pm5.32da  425  pm5.32  426  anbi2d  437  cbvex2  1794  cores  4767  isoini  5400  mpt2eq123  5506  genpassl  6507  genpassu  6508  fzind  8129  btwnz  8133  elfzm11  8723