Theorem excomw 2079

Description: Weak version of excom 2200 and biconditional form of excomimw 2077. Uses only Tarski's FOL axiom schemes. (Contributed by TM, 24-Jan-2026.)

Hypotheses

Ref	Expression
excomw.1	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
excomw.2	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
excomw	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Distinct variable groups:   𝜑,𝑧   𝜑,𝑤   𝜓,𝑥   𝜒,𝑦   𝑥,𝑦   𝑦,𝑧   𝑥,𝑤

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑦, 𝑧, 𝑤)   𝜒(𝑥, 𝑧, 𝑤)

Proof of Theorem excomw

Step	Hyp	Ref	Expression
1		excomw.1	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
2	1	excomimw 2077	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
3		excomw.2	. . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))
4	3	excomimw 2077	. 2 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)
5	2, 4	impbii 212	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  dm0rn0  5916  rnco  6255