Theorem pm5.32 488

Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125.

Assertion

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

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		pm4.11 400	. . . 4 ⊢ ((ψ ↔ χ) ↔ (¬ ψ ↔ ¬ χ))
2	1	imbi2i 160	. . 3 ⊢ ((φ → (ψ ↔ χ)) ↔ (φ → (¬ ψ ↔ ¬ χ)))
3		pm5.74 442	. . 3 ⊢ ((φ → (¬ ψ ↔ ¬ χ)) ↔ ((φ → ¬ ψ) ↔ (φ → ¬ χ)))
4		pm4.11 400	. . 3 ⊢ (((φ → ¬ ψ) ↔ (φ → ¬ χ)) ↔ (¬ (φ → ¬ ψ) ↔ ¬ (φ → ¬ χ)))
5	2, 3, 4	3bitr 155	. 2 ⊢ ((φ → (ψ ↔ χ)) ↔ (¬ (φ → ¬ ψ) ↔ ¬ (φ → ¬ χ)))
6		df-an 198	. . 3 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
7		df-an 198	. . 3 ⊢ ((φ ∧ χ) ↔ ¬ (φ → ¬ χ))
8	6, 7	bibi12i 462	. 2 ⊢ (((φ ∧ ψ) ↔ (φ ∧ χ)) ↔ (¬ (φ → ¬ ψ) ↔ ¬ (φ → ¬ χ)))
9	5, 8	bitr4 154	1 ⊢ ((φ → (ψ ↔ χ)) ↔ ((φ ∧ ψ) ↔ (φ ∧ χ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  pm5.32i 489  pm5.32d 491  cbval2 974  cbvex2 975

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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