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Metamath Proof Explorer

Theorem cbvexvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvexv for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 19-Apr-2017)

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

Proof

Step	Hyp	Ref	Expression
1		cbvalvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	cbvalvw	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ∀ 𝑦 ¬ 𝜓 )
4	3	notbii	⊢ ( ¬ ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∀ 𝑦 ¬ 𝜓 )
5		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
6		df-ex	⊢ ( ∃ 𝑦 𝜓 ↔ ¬ ∀ 𝑦 ¬ 𝜓 )
7	4 5 6	3bitr4i	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )