Theorem nfrexall-P8.RC 1088

Description: Inference form of nfrexall-P8 1086. †

Hypotheses

Ref	Expression
nfrexall-P8.RC.1	⊢ Ⅎ𝑥𝜑
nfrexall-P8.RC.2	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
nfrexall-P8.RC	⊢ ∀𝑥𝜑

Proof of Theorem nfrexall-P8.RC

Step	Hyp	Ref	Expression
1		nfrexall-P8.RC.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfrexall-P8.RC.2	. . . 4 ⊢ ∃𝑥𝜑
3	2	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → ∃𝑥𝜑)
4	1, 3	nfrexall-P8 1086	. 2 ⊢ (⊤ → ∀𝑥𝜑)
5	4	ndtruee-P3.18 183	1 ⊢ ∀𝑥𝜑

Syntax hints:  ∀wff-forall 8  ⊤wff-true 153  ∃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.RC.VR  1089