Theorem altxpeq1

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

	Ref	Expression
Assertion	altxpeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ×× 𝐶 ) = ( 𝐵 ×× 𝐶 ) )

Proof

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