Theorem pm2.61an2 365

Description: Elimination of an antecedent.

Hypotheses

Ref	Expression
pm2.61an2.1	⊢ ((φ ∧ ψ) → χ)
pm2.61an2.2	⊢ ((φ ∧ ¬ ψ) → χ)

Assertion

Ref	Expression
pm2.61an2	⊢ (φ → χ)

Proof of Theorem pm2.61an2

Step	Hyp	Ref	Expression
1		pm2.61an2.1	. . 3 ⊢ ((φ ∧ ψ) → χ)
2	1	exp 291	. 2 ⊢ (φ → (ψ → χ))
3		pm2.61an2.2	. . 3 ⊢ ((φ ∧ ¬ ψ) → χ)
4	3	exp 291	. 2 ⊢ (φ → (¬ ψ → χ))
5	2, 4	pm2.61d 112	1 ⊢ (φ → χ)

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

This theorem is referenced by:  opth2 1909  pw2en 3348  znnen 4930

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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