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Metamath Proof Explorer

Theorem cbvexdvaw

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017) Avoid ax-13 . (Revised by GG, 10-Jan-2024) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024)

		Ref	Expression
	Hypothesis	cbvaldvaw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
	Assertion	cbvexdvaw	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvaldvaw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	notbid	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
3	2	cbvaldvaw	⊢ ( 𝜑 → ( ∀ 𝑥 ¬ 𝜓 ↔ ∀ 𝑦 ¬ 𝜒 ) )
4		alnex	⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
5		alnex	⊢ ( ∀ 𝑦 ¬ 𝜒 ↔ ¬ ∃ 𝑦 𝜒 )
6	3 4 5	3bitr3g	⊢ ( 𝜑 → ( ¬ ∃ 𝑥 𝜓 ↔ ¬ ∃ 𝑦 𝜒 ) )
7	6	con4bid	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )