Theorem lemma-L5.05a 668

Description: A lemma for commuting existential quantifiers.

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

Hypothesis

Ref	Expression
lemma-L5.05a.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))

Assertion

Ref	Expression
lemma-L5.05a	⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

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

Proof of Theorem lemma-L5.05a

Step	Hyp	Ref	Expression
1		lemma-L5.05a.1	. . . . 5 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	subneg-P3.39 323	. . . 4 ⊢ (𝑥 = 𝑥₁ → (¬ 𝜑 ↔ ¬ 𝜑₁))
3	2	lemma-L5.04a 667	. . 3 ⊢ (∀𝑦∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)
4		lemma-L5.03a 666	. . 3 ⊢ (∀𝑦∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑦∃𝑥𝜑)
5		lemma-L5.03a 666	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑦𝜑)
6	3, 4, 5	subimd2-P4.RC 545	. 2 ⊢ (¬ ∃𝑦∃𝑥𝜑 → ¬ ∃𝑥∃𝑦𝜑)
7	6	trnsp-P3.31d.RC 289	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595

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:  excommw-P5  670  nfrex2w-P6  695