Theorem nfrgenw-P6 684

Description: ENF in ⇒ General for (weakened form).

Requires the existence of '𝜑₁(𝑥₁)' as a replacement for '𝜑(𝑥)'.

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

Hypotheses

Ref	Expression
nfrgenw-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
nfrgenw-P6.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrgenw-P6	⊢ (𝜑 → ∀𝑥𝜑)

Distinct variable groups:   𝜑,𝑥₁   𝜑₁,𝑥   𝑥,𝑥₁

Proof of Theorem nfrgenw-P6

Step	Hyp	Ref	Expression
1		nfrgenw-P6.1	. . 3 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	exiw-P5 662	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3		nfrgenw-P6.2	. . 3 ⊢ Ⅎ𝑥𝜑
4		dfnfreealt-P6 683	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	3, 4	bimpf-P4.RC 532	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
6	2, 5	syl-P3.24.RC 260	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃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

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:  nfrall2w-P6  694  nfrex2w-P6  695  nfrexgenw-P6  696  qremallw-P6  702