Theorem ectocld 8824

Description: Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 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		ectocld.3	. . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑)
2		ectocl.2	. . . . 5 ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	eqcoms 2745	. . . 4 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
4	1, 3	syl5ibcom 245	. . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
5	4	rexlimdva 3155	. 2 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓))
6		elqsi 8810	. . 3 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
7		ectocl.1	. . 3 ⊢ 𝑆 = (𝐵 / 𝑅)
8	6, 7	eleq2s 2859	. 2 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
9	5, 8	impel 505	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  [cec 8743   / cqs 8744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-qs 8751

This theorem is referenced by:  ectocl  8825  elqsn0  8826  qsdisj  8834  qsel  8836  eqgen  19199  orbsta  19331  sylow1lem3  19618  sylow2alem2  19636  sylow2a  19637  sylow2blem2  19639  frgpup1  19793  frgpup3lem  19795  quscrng  21293  pi1xfr  25088  pi1coghm  25094  vitalilem3  25645  qsdisjALTV  38616  eqvrelqsel  38617