Theorem exgennfr-P6 736

Description: Dual Form of gennfr-P6 734.

See exgennfrw-P6 697 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
exgennfr-P6.1	⊢ (∃𝑥𝜑 → 𝜑)

Assertion

Ref	Expression
exgennfr-P6	⊢ Ⅎ𝑥𝜑

Proof of Theorem exgennfr-P6

Step	Hyp	Ref	Expression
1		exgennfr-P6.1	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
2		lemma-L5.01a 600	. . . 4 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	bimpf-P4.RC 532	. . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
4	3	gennfr-P6 734	. 2 ⊢ Ⅎ𝑥 ¬ 𝜑
5		nfrneg-P6 688	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
6	4, 5	bimpf-P4.RC 532	1 ⊢ Ⅎ𝑥𝜑

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-L10 27

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