Theorem poe-P1.11a 65

Description: Principle of Explosion A.

A contradiction implies anything. The other form is poe-P1.11b 66.

Assertion

Ref	Expression
poe-P1.11a	⊢ (¬ 𝜑 → (𝜑 → 𝜓))

Proof of Theorem poe-P1.11a

Step	Hyp	Ref	Expression
1		ax-L1 11	. 2 ⊢ (¬ 𝜑 → (¬ 𝜓 → ¬ 𝜑))
2		ax-L3 13	. 2 ⊢ ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓))
3	1, 2	syl-P1.2 34	1 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem is referenced by:  poe-P1.11b  66  clav-P1.12  68  dneg-P1.13a  71