Theorem ectocl

Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995) (Revised by Mario Carneiro, 9-Jul-2014)

	Ref	Expression
Hypotheses	ectocl.1	⊢ 𝑆 = ( 𝐵 / 𝑅 )
	ectocl.2	⊢ ( [ 𝑥 ] 𝑅 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ectocl.3	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
Assertion	ectocl	⊢ ( 𝐴 ∈ 𝑆 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ectocl.1	⊢ 𝑆 = ( 𝐵 / 𝑅 )
2		ectocl.2	⊢ ( [ 𝑥 ] 𝑅 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		ectocl.3	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
4		tru	⊢ ⊤
5	3	adantl	⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐵 ) → 𝜑 )
6	1 2 5	ectocld	⊢ ( ( ⊤ ∧ 𝐴 ∈ 𝑆 ) → 𝜓 )
7	4 6	mpan	⊢ ( 𝐴 ∈ 𝑆 → 𝜓 )