Theorem qseq1

Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995)

	Ref	Expression
Assertion	qseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rexeq	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 = [ 𝑥 ] 𝐶 ) )
2	1	abbidv	⊢ ( 𝐴 = 𝐵 → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝐶 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 = [ 𝑥 ] 𝐶 } )
3		df-qs	⊢ ( 𝐴 / 𝐶 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝐶 }
4		df-qs	⊢ ( 𝐵 / 𝐶 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 = [ 𝑥 ] 𝐶 }
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) )