Theorem ectocl 5993

Description: Implicit substitution of class for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)

Hypotheses

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

Assertion

Ref	Expression
ectocl	⊢ (A ∈ S → ψ)

Distinct variable groups:   x,A   x,B   x,R   ψ,x

Allowed substitution hints:   φ(x)   S(x)

Proof of Theorem ectocl

Step	Hyp	Ref	Expression
1		tru 1321	. 2 ⊢ ⊤
2		ectocl.1	. . 3 ⊢ S = (B / R)
3		ectocl.2	. . 3 ⊢ ([x]R = A → (φ ↔ ψ))
4		ectocl.3	. . . 4 ⊢ (x ∈ B → φ)
5	4	adantl 452	. . 3 ⊢ (( ⊤ ∧ x ∈ B) → φ)
6	2, 3, 5	ectocld 5992	. 2 ⊢ (( ⊤ ∧ A ∈ S) → ψ)
7	1, 6	mpan 651	1 ⊢ (A ∈ S → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ⊤ wtru 1316   = wceq 1642   ∈ wcel 1710  [cec 5946   / cqs 5947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-qs 5952

This theorem is referenced by: (None)