Theorem dedlemb 570

Description: Lemma for weak deduction theorem.

Assertion

Ref	Expression
dedlemb	⊢ (¬ φ → (χ ↔ ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))

Proof of Theorem dedlemb

Step	Hyp	Ref	Expression
1		pm3.21 233	. . 3 ⊢ (¬ φ → (χ → (χ ∧ ¬ φ)))
2		olc 224	. . 3 ⊢ ((χ ∧ ¬ φ) → ((ψ ∧ φ) ∨ (χ ∧ ¬ φ)))
3	1, 2	syl6 23	. 2 ⊢ (¬ φ → (χ → ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))
4		pm2.21 71	. . . . 5 ⊢ (¬ φ → (φ → (ψ → χ)))
5	4	com23 32	. . . 4 ⊢ (¬ φ → (ψ → (φ → χ)))
6	5	imp3a 279	. . 3 ⊢ (¬ φ → ((ψ ∧ φ) → χ))
7		pm3.26 256	. . . 4 ⊢ ((χ ∧ ¬ φ) → χ)
8	7	a1i 7	. . 3 ⊢ (¬ φ → ((χ ∧ ¬ φ) → χ))
9	6, 8	jaod 329	. 2 ⊢ (¬ φ → (((ψ ∧ φ) ∨ (χ ∧ ¬ φ)) → χ))
10	3, 9	impbid 397	1 ⊢ (¬ φ → (χ ↔ ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  elimh 571  consensus 574  iffalse 1781

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-or 197  df-an 198

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