Theorem ectocld 6718

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 6704	. . . 4 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
2		ectocl.1	. . . 4 ⊢ 𝑆 = (𝐵 / 𝑅)
3	1, 2	eleq2s 2304	. . 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 2212	. . . . 5 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
7	4, 6	syl5ibcom 155	. . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
8	7	rexlimdva 2628	. . 3 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓))
9	3, 8	syl5 32	. 2 ⊢ (𝜒 → (𝐴 ∈ 𝑆 → 𝜓))
10	9	imp 124	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1375   ∈ wcel 2180  ∃wrex 2489  [cec 6648   / cqs 6649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-qs 6656

This theorem is referenced by:  ectocl  6719  elqsn0m  6720  qsel  6729  eqgen  13730  quscrng  14462