Theorem ectocld 8781

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 2771	. . . 4 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
4	1, 3	syl5ibcom 248	. . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
5	4	rexlimdva 3166	. 2 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓))
6		elqsi 8764	. . 3 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
7		ectocl.1	. . 3 ⊢ 𝑆 = (𝐵 / 𝑅)
8	6, 7	eleq2s 2881	. 2 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
9	5, 8	impel 514	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  [cec 8693   / cqs 8694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-qs 8701

This theorem is referenced by:  ectocl  8782  elqsn0  8783  qsdisj  8793  qsel  8795  eqgen  19250  orbsta  19384  sylow1lem3  19671  sylow2alem2  19689  sylow2a  19690  sylow2blem2  19692  frgpup1  19846  frgpup3lem  19848  quscrng  21404  pi1xfr  25195  pi1coghm  25201  vitalilem3  25750  qsdisjALTV  39326  eqvrelqsel  39327