Theorem excomw 2047

Description: Weak version of excom 2165 and biconditional form of excomimw 2045. 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 2045	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
3		excomw.2	. . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))
4	3	excomimw 2045	. 2 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)
5	2, 4	impbii 209	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  dm0rn0  5864  rnco  6199