Theorem ectocld 6398

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 6384	. . . 4 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅)
2		ectocl.1	. . . 4 ⊢ 𝑆 = (𝐵 / 𝑅)
3	1, 2	eleq2s 2189	. . 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 2098	. . . . 5 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓))
7	4, 6	syl5ibcom 154	. . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓))
8	7	rexlimdva 2502	. . 3 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓))
9	3, 8	syl5 32	. 2 ⊢ (𝜒 → (𝐴 ∈ 𝑆 → 𝜓))
10	9	imp 123	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∃wrex 2371  [cec 6330   / cqs 6331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-qs 6338

This theorem is referenced by:  ectocl  6399  elqsn0m  6400  qsel  6409