Theorem nfrgen-P6 733

Description: ENF in ⇒ General for.

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

See nfrgenw-P6 684 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
nfrgen-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrgen-P6	⊢ (𝜑 → ∀𝑥𝜑)

Proof of Theorem nfrgen-P6

Step	Hyp	Ref	Expression
1		exi-P6 718	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		nfrgen-P6.1	. . 3 ⊢ Ⅎ𝑥𝜑
3		dfnfreealt-P6 683	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	2, 3	bimpf-P4.RC 532	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	syl-P3.24.RC 260	1 ⊢ (𝜑 → ∀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-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

This theorem is referenced by:  nfrexgen-P6  735  genall-P6  737  nfrall2-P6  743  allic-P6  745