Theorem pm2.21dd 99

Description: A contradiction implies anything. Deduction from pm2.21 100. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses

Ref	Expression
pm2.21dd.1	⊢ (φ → ψ)
pm2.21dd.2	⊢ (φ → ¬ ψ)

Assertion

Ref	Expression
pm2.21dd	⊢ (φ → χ)

Proof of Theorem pm2.21dd

Step	Hyp	Ref	Expression
1		pm2.21dd.1	. 2 ⊢ (φ → ψ)
2		pm2.21dd.2	. . 3 ⊢ (φ → ¬ ψ)
3	2	pm2.21d 98	. 2 ⊢ (φ → (ψ → χ))
4	1, 3	mpd 14	1 ⊢ (φ → χ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  pm2.21fal  1335  pm2.21ddne  2591