Theorem allcommw-P5 669

Description: Universal Quantifier Commutivity (weakened form).

Requires the existence of '𝜑₁(𝑥₁)' as a replacement for '𝜑(𝑥)', and the existance of '𝜑₂(𝑦₁)' as a replacement for '𝜑(𝑦)'.

The form not requiring any hypotheses, but relying on an additional axiom is allcomm-P6 739.

Hypotheses

Ref	Expression
allcommw-P5.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
allcommw-P5.2	⊢ (𝑦 = 𝑦₁ → (𝜑 ↔ 𝜑₂))

Assertion

Ref	Expression
allcommw-P5	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

Distinct variable groups:   𝜑,𝑥₁   𝜑₁,𝑥   𝜑,𝑦₁   𝜑₂,𝑦   𝑥,𝑦,𝑥₁,𝑦₁

Proof of Theorem allcommw-P5

Step	Hyp	Ref	Expression
1		allcommw-P5.2	. . 3 ⊢ (𝑦 = 𝑦₁ → (𝜑 ↔ 𝜑₂))
2	1	lemma-L5.04a 667	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		allcommw-P5.1	. . 3 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
4	3	lemma-L5.04a 667	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
5	2, 4	rcp-NDBII0 239	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: (None)