Theorem cbvixpdavw

Description: Change bound variable in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvixpdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
Assertion	cbvixpdavw	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑦 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		cbvixpdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
2		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	3	cbvabdavw	⊢ ( 𝜑 → { 𝑥 ∣ 𝑥 ∈ 𝐴 } = { 𝑦 ∣ 𝑦 ∈ 𝐴 } )
5	4	fneq2d	⊢ ( 𝜑 → ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ↔ 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐴 } ) )
6		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝑥 = 𝑦 )
7	6	fveq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ‘ 𝑥 ) = ( 𝑡 ‘ 𝑦 ) )
8	7 1	eleq12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑡 ‘ 𝑥 ) ∈ 𝐵 ↔ ( 𝑡 ‘ 𝑦 ) ∈ 𝐶 ) )
9	8	cbvraldva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐵 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑡 ‘ 𝑦 ) ∈ 𝐶 ) )
10	5 9	anbi12d	⊢ ( 𝜑 → ( ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐵 ) ↔ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐴 } ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑡 ‘ 𝑦 ) ∈ 𝐶 ) ) )
11	10	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐵 ) } = { 𝑡 ∣ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐴 } ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑡 ‘ 𝑦 ) ∈ 𝐶 ) } )
12		df-ixp	⊢ X 𝑥 ∈ 𝐴 𝐵 = { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐵 ) }
13		df-ixp	⊢ X 𝑦 ∈ 𝐴 𝐶 = { 𝑡 ∣ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐴 } ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑡 ‘ 𝑦 ) ∈ 𝐶 ) }
14	11 12 13	3eqtr4g	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 = X 𝑦 ∈ 𝐴 𝐶 )