Theorem nfrall2w-P6 694

Description: ENF Over Universal Quantifier (different variable - weakened form).

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

Hypotheses

Ref	Expression
nfrall2w-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
nfrall2w-P6.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrall2w-P6	⊢ Ⅎ𝑥∀𝑦𝜑

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

Proof of Theorem nfrall2w-P6

Step	Hyp	Ref	Expression
1		nfrall2w-P6.1	. . 3 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	suballv-P5 623	. 2 ⊢ (𝑥 = 𝑥₁ → (∀𝑦𝜑 ↔ ∀𝑦𝜑₁))
3		nfrall2w-P6.2	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	1, 3	nfrgenw-P6 684	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
5	4	alloverim-P5.RC.GEN 592	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
6	1	lemma-L5.04a 667	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
7	5, 6	syl-P3.24.RC 260	. 2 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
8	2, 7	gennfrw-P6 685	1 ⊢ Ⅎ𝑥∀𝑦𝜑

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  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: (None)