Theorem pm5.75 903

Description: Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)

Assertion

Ref	Expression
pm5.75	⊢ (((χ → ¬ ψ) ∧ (φ ↔ (ψ ∨ χ))) → ((φ ∧ ¬ ψ) ↔ χ))

Proof of Theorem pm5.75

Step	Hyp	Ref	Expression
1		anbi1 687	. . 3 ⊢ ((φ ↔ (ψ ∨ χ)) → ((φ ∧ ¬ ψ) ↔ ((ψ ∨ χ) ∧ ¬ ψ)))
2		orcom 376	. . . . 5 ⊢ ((ψ ∨ χ) ↔ (χ ∨ ψ))
3	2	anbi1i 676	. . . 4 ⊢ (((ψ ∨ χ) ∧ ¬ ψ) ↔ ((χ ∨ ψ) ∧ ¬ ψ))
4		pm5.61 693	. . . 4 ⊢ (((χ ∨ ψ) ∧ ¬ ψ) ↔ (χ ∧ ¬ ψ))
5	3, 4	bitri 240	. . 3 ⊢ (((ψ ∨ χ) ∧ ¬ ψ) ↔ (χ ∧ ¬ ψ))
6	1, 5	syl6bb 252	. 2 ⊢ ((φ ↔ (ψ ∨ χ)) → ((φ ∧ ¬ ψ) ↔ (χ ∧ ¬ ψ)))
7		pm4.71 611	. . . 4 ⊢ ((χ → ¬ ψ) ↔ (χ ↔ (χ ∧ ¬ ψ)))
8	7	biimpi 186	. . 3 ⊢ ((χ → ¬ ψ) → (χ ↔ (χ ∧ ¬ ψ)))
9	8	bicomd 192	. 2 ⊢ ((χ → ¬ ψ) → ((χ ∧ ¬ ψ) ↔ χ))
10	6, 9	sylan9bbr 681	1 ⊢ (((χ → ¬ ψ) ∧ (φ ↔ (ψ ∨ χ))) → ((φ ∧ ¬ ψ) ↔ χ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ 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-or 359  df-an 360

This theorem is referenced by: (None)