Theorem nfrexall-P8 1086

Description: Combining of nfrgen-P8 1076 and nfrexgen-P8 1080. †

Hypotheses

Ref	Expression
nfrexall-P8.1	⊢ Ⅎ𝑥𝜑
nfrexall-P8.2	⊢ (𝛾 → ∃𝑥𝜑)

Assertion

Ref	Expression
nfrexall-P8	⊢ (𝛾 → ∀𝑥𝜑)

Proof of Theorem nfrexall-P8

Step	Hyp	Ref	Expression
1		nfrexall-P8.2	. 2 ⊢ (𝛾 → ∃𝑥𝜑)
2		nfrexall-P8.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfrexgen-P7.CL 932	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
4	2	nfrgen-P7.CL 930	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
5	1, 3, 4	dsyl-P3.25.RC 262	1 ⊢ (𝛾 → ∀𝑥𝜑)

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:  nfrexall-P8.VR  1087  nfrexall-P8.RC  1088  nfrexall-P8.CL  1090