Theorem exallnfr-P8 1092

Description: Converse of nfrexall-P8.CL 1090. †

Hypothesis

Ref	Expression
exallnfr-P8.1	⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
exallnfr-P8	⊢ Ⅎ𝑥𝜑

Proof of Theorem exallnfr-P8

Step	Hyp	Ref	Expression
1		exi-P7.CL 952	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2		exallnfr-P8.1	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
3	1, 2	syl-P3.24.RC 260	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	gennfr-P8 1079	1 ⊢ Ⅎ𝑥𝜑

Syntax hints:  ∀wff-forall 8   → 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-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by: (None)