Theorem ectocld 6172

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		elqsi 6158	. . . 4 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
2		ectocl.1	. . . 4 ⊢ 𝑆 = (𝐵 / 𝑅)
3	1, 2	eleq2s 2132	. . 3 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
4		ectocld.3	. . . . 5 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑)
5		ectocl.2	. . . . . 6 ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	eqcoms 2043	. . . . 5 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
7	4, 6	syl5ibcom 144	. . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
8	7	rexlimdva 2433	. . 3 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓))
9	3, 8	syl5 28	. 2 ⊢ (𝜒 → (𝐴 ∈ 𝑆 → 𝜓))
10	9	imp 115	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  [cec 6104   / cqs 6105

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-qs 6112

This theorem is referenced by:  ectocl  6173  elqsn0m  6174  qsel  6183