Theorem ectocld 8759

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 2769	. . . 4 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
4	1, 3	syl5ibcom 247	. . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
5	4	rexlimdva 3162	. 2 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓))
6		elqsi 8742	. . 3 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
7		ectocl.1	. . 3 ⊢ 𝑆 = (𝐵 / 𝑅)
8	6, 7	eleq2s 2879	. 2 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
9	5, 8	impel 513	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  [cec 8671   / cqs 8672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3086  df-qs 8679

This theorem is referenced by:  ectocl  8760  elqsn0  8761  qsdisj  8771  qsel  8773  eqgen  19205  orbsta  19336  sylow1lem3  19623  sylow2alem2  19641  sylow2a  19642  sylow2blem2  19644  frgpup1  19798  frgpup3lem  19800  quscrng  21333  pi1xfr  25097  pi1coghm  25103  vitalilem3  25652  qsdisjALTV  39162  eqvrelqsel  39163