Theorem ectocld 8778

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 2741	. . . 4 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
4	1, 3	syl5ibcom 244	. . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
5	4	rexlimdva 3156	. 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 2852	. 2 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
9	5, 8	impel 507	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  [cec 8701   / cqs 8702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-qs 8709

This theorem is referenced by:  ectocl  8779  elqsn0  8780  qsdisj  8788  qsel  8790  eqgen  19061  orbsta  19177  sylow1lem3  19468  sylow2alem2  19486  sylow2a  19487  sylow2blem2  19489  frgpup1  19643  frgpup3lem  19645  quscrng  20878  pi1xfr  24571  pi1coghm  24577  vitalilem3  25127  qsdisjALTV  37485  eqvrelqsel  37486