Theorem ectocl 8782

Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

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

Assertion

Ref	Expression
ectocl	⊢ (𝐴 ∈ 𝑆 → 𝜓)

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

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

Proof of Theorem ectocl

Step	Hyp	Ref	Expression
1		tru 1574	. 2 ⊢ ⊤
2		ectocl.1	. . 3 ⊢ 𝑆 = (𝐵 / 𝑅)
3		ectocl.2	. . 3 ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))
4		ectocl.3	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
5	4	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑)
6	2, 3, 5	ectocld 8781	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓)
7	1, 6	mpan 702	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  [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:  vitalilem2  25749