Theorem pm5.74 442

Description: Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126.

Assertion

Ref	Expression
pm5.74	⊢ ((φ → (ψ ↔ χ)) ↔ ((φ → ψ) ↔ (φ → χ)))

Proof of Theorem pm5.74

Step	Hyp	Ref	Expression
1		bi1 130	. . . . 5 ⊢ ((ψ ↔ χ) → (ψ → χ))
2	1	syl3 18	. . . 4 ⊢ ((φ → (ψ ↔ χ)) → (φ → (ψ → χ)))
3	2	a2d 15	. . 3 ⊢ ((φ → (ψ ↔ χ)) → ((φ → ψ) → (φ → χ)))
4		bi2 131	. . . . 5 ⊢ ((ψ ↔ χ) → (χ → ψ))
5	4	syl3 18	. . . 4 ⊢ ((φ → (ψ ↔ χ)) → (φ → (χ → ψ)))
6	5	a2d 15	. . 3 ⊢ ((φ → (ψ ↔ χ)) → ((φ → χ) → (φ → ψ)))
7	3, 6	impbid 397	. 2 ⊢ ((φ → (ψ ↔ χ)) → ((φ → ψ) ↔ (φ → χ)))
8		bi1 130	. . . . 5 ⊢ (((φ → ψ) ↔ (φ → χ)) → ((φ → ψ) → (φ → χ)))
9	8	pm2.86d 65	. . . 4 ⊢ (((φ → ψ) ↔ (φ → χ)) → (φ → (ψ → χ)))
10		bi2 131	. . . . 5 ⊢ (((φ → ψ) ↔ (φ → χ)) → ((φ → χ) → (φ → ψ)))
11	10	pm2.86d 65	. . . 4 ⊢ (((φ → ψ) ↔ (φ → χ)) → (φ → (χ → ψ)))
12	9, 11	anim12d 431	. . 3 ⊢ (((φ → ψ) ↔ (φ → χ)) → ((φ ∧ φ) → ((ψ → χ) ∧ (χ → ψ))))
13		anidm 331	. . . 4 ⊢ ((φ ∧ φ) ↔ φ)
14	13	bicomi 150	. . 3 ⊢ (φ ↔ (φ ∧ φ))
15		bi 396	. . 3 ⊢ ((ψ ↔ χ) ↔ ((ψ → χ) ∧ (χ → ψ)))
16	12, 14, 15	3imtr4g 426	. 2 ⊢ (((φ → ψ) ↔ (φ → χ)) → (φ → (ψ ↔ χ)))
17	7, 16	impbi 139	1 ⊢ ((φ → (ψ ↔ χ)) ↔ ((φ → ψ) ↔ (φ → χ)))

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

This theorem is referenced by:  pm5.74i 443  pm5.74d 444  pm5.74ri 445  pm5.74rd 446  pm5.32 488

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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