Theorem nfrcond-P8 1108

Description: ENF in Antecedent ⇒ Conditionally Not Free in Consequent. †

This is actually a generalization of nfrthm-P8 1107.

Hypotheses

Ref	Expression
nfrcond-P8.1	⊢ Ⅎ𝑥𝛾
nfrcond-P8.2	⊢ (𝛾 → 𝜑)

Assertion

Ref	Expression
nfrcond-P8	⊢ (𝛾 → Ⅎ𝑥𝜑)

Proof of Theorem nfrcond-P8

Step	Hyp	Ref	Expression
1		nfrcond-P8.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		nfrcond-P8.2	. . . . 5 ⊢ (𝛾 → 𝜑)
3	1, 2	alli-P7 947	. . . 4 ⊢ (𝛾 → ∀𝑥𝜑)
4	3	axL1-P3.21 252	. . 3 ⊢ (𝛾 → (𝜑 → ∀𝑥𝜑))
5	1, 4	alli-P7 947	. 2 ⊢ (𝛾 → ∀𝑥(𝜑 → ∀𝑥𝜑))
6		gennfrcl-P7 963	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
7	5, 6	syl-P3.24.RC 260	1 ⊢ (𝛾 → Ⅎ𝑥𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  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:  nfrcond-P8.VR  1109