Theorem cbvexd

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvexdw if possible. (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvald.1	⊢ Ⅎ 𝑦 𝜑
2		cbvald.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3		cbvald.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	cbvald	⊢ ( 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 ) )