Theorem nfrexgen-P6 735

Description: Dual Form of nfrgen-P6 733.

See nfrexgenw-P6 696 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
nfrexgen-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrexgen-P6	⊢ (∃𝑥𝜑 → 𝜑)

Proof of Theorem nfrexgen-P6

Step	Hyp	Ref	Expression
1		nfrexgen-P6.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfrneg-P6 688	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
3	1, 2	bimpr-P4.RC 534	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	3	nfrgen-P6 733	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
5		lemma-L5.01a 600	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
6	4, 5	bimpr-P4.RC 534	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10  ∃wff-exists 595  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L12 29

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  df-nfree-D6.1 682

This theorem is referenced by:  nfrex2-P6  744  exia-P6  746  lemma-L6.04a  749  trnsvsub-P6  763