Theorem cbvrexdva2

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 12-Aug-2023)

	Ref	Expression
Hypotheses	cbvraldva2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
	cbvraldva2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvrexdva2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvraldva2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		cbvraldva2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝑥 = 𝑦 )
4	3 2	eleq12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	4 1	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
6	5	ancoms	⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
7	6	pm5.32da	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) ) )
8	7	cbvexvw	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ↔ ∃ 𝑦 ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
9		19.42v	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
10		19.42v	⊢ ( ∃ 𝑦 ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) ↔ ( 𝜑 ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
11	8 9 10	3bitr3i	⊢ ( ( 𝜑 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
12		pm5.32	⊢ ( ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) ) ↔ ( ( 𝜑 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) ) )
13	11 12	mpbir	⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) ) )
14		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
15		df-rex	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜒 ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜒 ) )
16	13 14 15	3bitr4g	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )