Theorem axL5ex-P5 613

Description: Dual of ax-L5 17.

'𝑥' does not occur in '𝜑'.

Assertion

Ref	Expression
axL5ex-P5	⊢ (∃𝑥𝜑 → 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem axL5ex-P5

Step	Hyp	Ref	Expression
1		ax-L5 17	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		lemma-L5.01a 600	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	bimpr-P4.RC 534	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-L5 17

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  exiav-P5  615  qimeqallav-P5-L1  617  qimeqallbv-P5-L1  619  lemma-L5.02a  653  qremexv-P5  657