Theorem gennfr-P8 1079

Description: General For ⇒ ENF. †

Hypothesis

Ref	Expression
gennfr-P8.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
gennfr-P8	⊢ Ⅎ𝑥𝜑

Proof of Theorem gennfr-P8

Step	Hyp	Ref	Expression
1		ndnfrall1-P7.7 832	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2		alle-P7.CL 942	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
3		gennfr-P8.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4	2, 3	rcp-NDBII0 239	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
5	4	ndnfrleq-P7.11.RC 882	. 2 ⊢ (Ⅎ𝑥∀𝑥𝜑 ↔ Ⅎ𝑥𝜑)
6	1, 5	bimpf-P4.RC 532	1 ⊢ 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:  exallnfr-P8  1092