Theorem ectocld

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

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

Proof

Step	Hyp	Ref	Expression
1		ectocl.1	⊢ 𝑆 = ( 𝐵 / 𝑅 )
2		ectocl.2	⊢ ( [ 𝑥 ] 𝑅 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		ectocld.3	⊢ ( ( 𝜒 ∧ 𝑥 ∈ 𝐵 ) → 𝜑 )
4	2	eqcoms	⊢ ( 𝐴 = [ 𝑥 ] 𝑅 → ( 𝜑 ↔ 𝜓 ) )
5	3 4	syl5ibcom	⊢ ( ( 𝜒 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐴 = [ 𝑥 ] 𝑅 → 𝜓 ) )
6	5	rexlimdva	⊢ ( 𝜒 → ( ∃ 𝑥 ∈ 𝐵 𝐴 = [ 𝑥 ] 𝑅 → 𝜓 ) )
7		elqsi	⊢ ( 𝐴 ∈ ( 𝐵 / 𝑅 ) → ∃ 𝑥 ∈ 𝐵 𝐴 = [ 𝑥 ] 𝑅 )
8	7 1	eleq2s	⊢ ( 𝐴 ∈ 𝑆 → ∃ 𝑥 ∈ 𝐵 𝐴 = [ 𝑥 ] 𝑅 )
9	6 8	impel	⊢ ( ( 𝜒 ∧ 𝐴 ∈ 𝑆 ) → 𝜓 )