Theorem excomw

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

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

Proof

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