Theorem pm5.32 617

Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)

Assertion

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

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		notbi 286	. . . 4 ⊢ ((ψ ↔ χ) ↔ (¬ ψ ↔ ¬ χ))
2	1	imbi2i 303	. . 3 ⊢ ((φ → (ψ ↔ χ)) ↔ (φ → (¬ ψ ↔ ¬ χ)))
3		pm5.74 235	. . 3 ⊢ ((φ → (¬ ψ ↔ ¬ χ)) ↔ ((φ → ¬ ψ) ↔ (φ → ¬ χ)))
4		notbi 286	. . 3 ⊢ (((φ → ¬ ψ) ↔ (φ → ¬ χ)) ↔ (¬ (φ → ¬ ψ) ↔ ¬ (φ → ¬ χ)))
5	2, 3, 4	3bitri 262	. 2 ⊢ ((φ → (ψ ↔ χ)) ↔ (¬ (φ → ¬ ψ) ↔ ¬ (φ → ¬ χ)))
6		df-an 360	. . 3 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
7		df-an 360	. . 3 ⊢ ((φ ∧ χ) ↔ ¬ (φ → ¬ χ))
8	6, 7	bibi12i 306	. 2 ⊢ (((φ ∧ ψ) ↔ (φ ∧ χ)) ↔ (¬ (φ → ¬ ψ) ↔ ¬ (φ → ¬ χ)))
9	5, 8	bitr4i 243	1 ⊢ ((φ → (ψ ↔ χ)) ↔ ((φ ∧ ψ) ↔ (φ ∧ χ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  pm5.32i  618  pm5.32d  620  xordi  865  cbval2  2004  cbvex2  2005  rabbi  2790