Theorem cbvexdw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvaldw.1	⊢ Ⅎ 𝑦 𝜑
	cbvaldw.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	cbvaldw.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	cbvexdw	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvaldw.1	⊢ Ⅎ 𝑦 𝜑
2		cbvaldw.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3		cbvaldw.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4	2	nfnd	⊢ ( 𝜑 → Ⅎ 𝑦 ¬ 𝜓 )
5		notbi	⊢ ( ( 𝜓 ↔ 𝜒 ) ↔ ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
6	3 5	syl6ib	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) )
7	1 4 6	cbvaldw	⊢ ( 𝜑 → ( ∀ 𝑥 ¬ 𝜓 ↔ ∀ 𝑦 ¬ 𝜒 ) )
8		alnex	⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
9		alnex	⊢ ( ∀ 𝑦 ¬ 𝜒 ↔ ¬ ∃ 𝑦 𝜒 )
10	7 8 9	3bitr3g	⊢ ( 𝜑 → ( ¬ ∃ 𝑥 𝜓 ↔ ¬ ∃ 𝑦 𝜒 ) )
11	10	con4bid	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )