Theorem ectocl 6568

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 1347	. 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 275	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑)
6	2, 3, 5	ectocld 6567	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓)
7	1, 6	mpan 421	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  ⊤wtru 1344   ∈ wcel 2136  [cec 6499   / cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-qs 6507

This theorem is referenced by: (None)