Theorem altxpeq2

Description: Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012)

	Ref	Expression
Assertion	altxpeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ×× 𝐴 ) = ( 𝐶 ×× 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rexeq	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑦 ∈ 𝐴 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ ↔ ∃ 𝑦 ∈ 𝐵 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ ) )
2	1	rexbidv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐴 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ ↔ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐵 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ ) )
3	2	abbidv	⊢ ( 𝐴 = 𝐵 → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐴 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ } = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐵 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ } )
4		df-altxp	⊢ ( 𝐶 ×× 𝐴 ) = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐴 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ }
5		df-altxp	⊢ ( 𝐶 ×× 𝐵 ) = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐵 𝑧 = ⟪ 𝑥 , 𝑦 ⟫ }
6	3 4 5	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ×× 𝐴 ) = ( 𝐶 ×× 𝐵 ) )