Theorem ectocld 5992

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

Hypotheses

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

Assertion

Ref	Expression
ectocld	⊢ ((χ ∧ A ∈ S) → ψ)

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

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

Proof of Theorem ectocld

Step	Hyp	Ref	Expression
1		elqsi 5979	. . . 4 ⊢ (A ∈ (B / R) → ∃x ∈ B A = [x]R)
2		ectocl.1	. . . 4 ⊢ S = (B / R)
3	1, 2	eleq2s 2445	. . 3 ⊢ (A ∈ S → ∃x ∈ B A = [x]R)
4		ectocld.3	. . . . 5 ⊢ ((χ ∧ x ∈ B) → φ)
5		ectocl.2	. . . . . 6 ⊢ ([x]R = A → (φ ↔ ψ))
6	5	eqcoms 2356	. . . . 5 ⊢ (A = [x]R → (φ ↔ ψ))
7	4, 6	syl5ibcom 211	. . . 4 ⊢ ((χ ∧ x ∈ B) → (A = [x]R → ψ))
8	7	rexlimdva 2739	. . 3 ⊢ (χ → (∃x ∈ B A = [x]R → ψ))
9	3, 8	syl5 28	. 2 ⊢ (χ → (A ∈ S → ψ))
10	9	imp 418	1 ⊢ ((χ ∧ A ∈ S) → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  [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:  ectocl  5993  elqsn0  5994  qsdisj  5996