Theorem nfrall2d-P6 819

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

Hypotheses

Ref	Expression
nfrall2d-P6.1	⊢ Ⅎ𝑥𝛾
nfrall2d-P6.2	⊢ Ⅎ𝑦𝛾
nfrall2d-P6.3	⊢ (𝛾 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
nfrall2d-P6	⊢ (𝛾 → Ⅎ𝑥∀𝑦𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem nfrall2d-P6

Step	Hyp	Ref	Expression
1		nfrall2d-P6.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		nfrall2d-P6.2	. . . . 5 ⊢ Ⅎ𝑦𝛾
3		nfrall2d-P6.3	. . . . . 6 ⊢ (𝛾 → Ⅎ𝑥𝜑)
4		nfrgencl-L6 811	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5	3, 4	syl-P3.24.RC 260	. . . . 5 ⊢ (𝛾 → (𝜑 → ∀𝑥𝜑))
6	2, 5	alloverim-P5.GENF 747	. . . 4 ⊢ (𝛾 → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
7		allcomm-P6 739	. . . . 5 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
8	7	subimr-P3.40b.RC 328	. . . 4 ⊢ ((∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) ↔ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
9	6, 8	subimr2-P4.RC 543	. . 3 ⊢ (𝛾 → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
10	1, 9	allic-P6 745	. 2 ⊢ (𝛾 → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
11		gennfrcl-L6 812	. 2 ⊢ (∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) → Ⅎ𝑥∀𝑦𝜑)
12	10, 11	syl-P3.24.RC 260	1 ⊢ (𝛾 → Ⅎ𝑥∀𝑦𝜑)

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

This theorem is referenced by:  ndnfrall2-P7.9  834