Theorem ectocl 7702

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 1479	. 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 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑)
6	2, 3, 5	ectocld 7701	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓)
7	1, 6	mpan 702	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  [cec 7627   / cqs 7628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-qs 7635

This theorem is referenced by:  vitalilem2  23184