Theorem gennfr-P6 734

Description: General for ⇒ ENF in.

If '𝜑' is general for '𝑥', then '𝑥' is effectively not free in '𝜑'.

See gennfrw-P6 685 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
gennfr-P6.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
gennfr-P6	⊢ Ⅎ𝑥𝜑

Proof of Theorem gennfr-P6

Step	Hyp	Ref	Expression
1		gennfr-P6.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	alloverimex-P5.RC.GEN 603	. . 3 ⊢ (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑)
3		exgenall-P6 732	. . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
4	2, 3	syl-P3.24.RC 260	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
5		dfnfreealt-P6 683	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6	4, 5	bimpr-P4.RC 534	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-L10 27

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

This theorem is referenced by:  exgennfr-P6  736  genall-P6  737  nfrall1-P6  741  nfrex1-P6  742  nfrall2-P6  743