Theorem ixpeq12dv

Description: Equality theorem for infinite Cartesian product. Deduction version. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	ixpeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	ixpeq12dv.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	ixpeq12dv	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		ixpeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ixpeq12dv.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3	abbidv	⊢ ( 𝜑 → { 𝑥 ∣ 𝑥 ∈ 𝐴 } = { 𝑥 ∣ 𝑥 ∈ 𝐵 } )
5	4	fneq2d	⊢ ( 𝜑 → ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ↔ 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ) )
6	3	imbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ) )
7	6	albidv	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ) )
8		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) )
9		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) )
10	7 8 9	3bitr4g	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) )
11	5 10	anbi12d	⊢ ( 𝜑 → ( ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ) )
12	11	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) } = { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) } )
13		df-ixp	⊢ X 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) }
14		df-ixp	⊢ X 𝑥 ∈ 𝐵 𝐶 = { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐵 } ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) }
15	12 13 14	3eqtr4g	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐶 )
16	2	ixpeq2dv	⊢ ( 𝜑 → X 𝑥 ∈ 𝐵 𝐶 = X 𝑥 ∈ 𝐵 𝐷 )
17	15 16	eqtrd	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑥 ∈ 𝐵 𝐷 )