Theorem ectocl 8365

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 1541	. 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 484	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑)
6	2, 3, 5	ectocld 8364	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓)
7	1, 6	mpan 688	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537  ⊤wtru 1538   ∈ wcel 2114  [cec 8287   / cqs 8288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-qs 8295

This theorem is referenced by:  vitalilem2  24210