Theorem nfrnegconv-P8 1110

Description: Converse of ndnfrneg-P7.2 827.

This statement is not deducible with intuitionist logic.

Hypothesis

Ref	Expression
nfrnegconv-P8.1	⊢ (𝛾 → Ⅎ𝑥 ¬ 𝜑)

Assertion

Ref	Expression
nfrnegconv-P8	⊢ (𝛾 → Ⅎ𝑥𝜑)

Proof of Theorem nfrnegconv-P8

Step	Hyp	Ref	Expression
1		nfrnegconv-P8.1	. 2 ⊢ (𝛾 → Ⅎ𝑥 ¬ 𝜑)
2		exnegall-P7 1046	. . . . . 6 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
3	2	rcp-NDBIER0 241	. . . . 5 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥 ¬ 𝜑)
4	3	rcp-NDIMP0addall 207	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (¬ ∀𝑥𝜑 → ∃𝑥 ¬ 𝜑))
5		dfnfreealtonlyif-P7 966	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (∃𝑥 ¬ 𝜑 → ∀𝑥 ¬ 𝜑))
6		allnegex-P7 958	. . . . . 6 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
7	6	rcp-NDBIEF0 240	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
8	7	rcp-NDIMP0addall 207	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑))
9	4, 5, 8	dsyl-P3.25 261	. . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (¬ ∀𝑥𝜑 → ¬ ∃𝑥𝜑))
10	9	trnsp-P3.31d 288	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
11		dfnfreealtif-P7 964	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
12	1, 10, 11	dsyl-P3.25.RC 262	1 ⊢ (𝛾 → 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-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  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-false-D2.5 158  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  nfrnegconv-P8.RC  1111  nfrnegconv-P8.CL  1112