Theorem excomimw

Description: Weak version of excomim . Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 23-Jun-2025)

	Ref	Expression
Hypothesis	excomimw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	excomimw	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑦 ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		excomimw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( 𝑥 = 𝑧 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	alcomimw	⊢ ( ∀ 𝑦 ∀ 𝑥 ¬ 𝜑 → ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )
4	3	con3i	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 → ¬ ∀ 𝑦 ∀ 𝑥 ¬ 𝜑 )
5		2exnaln	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ¬ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )
6		2exnaln	⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑦 ∀ 𝑥 ¬ 𝜑 )
7	4 5 6	3imtr4i	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑦 ∃ 𝑥 𝜑 )