Theorem lemma-L5.04a 667

Description: A lemma for commuting universal quantifiers.

Requires the existence of '𝜑₁(𝑦₁)' as a replacement for '𝜑(𝑦)'.

Hypothesis

Ref	Expression
lemma-L5.04a.1	⊢ (𝑦 = 𝑦₁ → (𝜑 ↔ 𝜑₁))

Assertion

Ref	Expression
lemma-L5.04a	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Distinct variable groups:   𝜑,𝑦₁   𝜑₁,𝑦   𝑥,𝑦,𝑦₁

Proof of Theorem lemma-L5.04a

Step	Hyp	Ref	Expression
1		lemma-L5.04a.1	. . . . 5 ⊢ (𝑦 = 𝑦₁ → (𝜑 ↔ 𝜑₁))
2	1	cbvallv-P5 659	. . . 4 ⊢ (∀𝑦𝜑 ↔ ∀𝑦₁𝜑₁)
3	2	rcp-NDBIEF0 240	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑦₁𝜑₁)
4	3	alloverim-P5.RC.GEN 592	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦₁𝜑₁)
5		ax-L5 17	. 2 ⊢ (∀𝑥∀𝑦₁𝜑₁ → ∀𝑦∀𝑥∀𝑦₁𝜑₁)
6	1	bisym-P3.33b 298	. . . . . 6 ⊢ (𝑦 = 𝑦₁ → (𝜑₁ ↔ 𝜑))
7		eqsym-P5.CL.SYM 629	. . . . . 6 ⊢ (𝑦 = 𝑦₁ ↔ 𝑦₁ = 𝑦)
8	6, 7	subiml2-P4.RC 541	. . . . 5 ⊢ (𝑦₁ = 𝑦 → (𝜑₁ ↔ 𝜑))
9	8	specisub-P5 654	. . . 4 ⊢ (∀𝑦₁𝜑₁ → 𝜑)
10	9	alloverim-P5.RC.GEN 592	. . 3 ⊢ (∀𝑥∀𝑦₁𝜑₁ → ∀𝑥𝜑)
11	10	alloverim-P5.RC.GEN 592	. 2 ⊢ (∀𝑦∀𝑥∀𝑦₁𝜑₁ → ∀𝑦∀𝑥𝜑)
12	4, 5, 11	dsyl-P3.25.RC 262	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

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

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

This theorem is referenced by:  lemma-L5.05a  668  allcommw-P5  669  nfrall2w-P6  694