Theorem cdleme31so 41216

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)

Hypotheses

Ref	Expression
cdleme31so.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
cdleme31so.c	⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))

Assertion

Ref	Expression
cdleme31so	⊢ (𝑋 ∈ 𝐵 → ⦋𝑋 / 𝑥⦌𝑂 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥, ∨   𝑥, ≤   𝑥, ∧   𝑥,𝑁   𝑥,𝑠,𝑧,𝑋   𝑥,𝑊

Allowed substitution hints:   𝐴(𝑧, 𝑠)   𝐵(𝑧, 𝑠)   𝐶(𝑥, 𝑧, 𝑠)   ∨ (𝑧, 𝑠)   ≤ (𝑧, 𝑠)   ∧ (𝑧, 𝑠)   𝑁(𝑧, 𝑠)   𝑂(𝑥, 𝑧, 𝑠)   𝑊(𝑧, 𝑠)

Proof of Theorem cdleme31so

Step	Hyp	Ref	Expression
1		nfcvd 2928	. . 3 ⊢ (𝑋 ∈ 𝐵 → Ⅎ𝑥(℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
2		oveq1 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑊) = (𝑋 ∧ 𝑊))
3	2	oveq2d 7436	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑠 ∨ (𝑥 ∧ 𝑊)) = (𝑠 ∨ (𝑋 ∧ 𝑊)))
4		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
5	3, 4	eqeq12d 2781	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥 ↔ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
6	5	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
7	2	oveq2d 7436	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑁 ∨ (𝑥 ∧ 𝑊)) = (𝑁 ∨ (𝑋 ∧ 𝑊)))
8	7	eqeq2d 2776	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)) ↔ 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))
9	6, 8	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑋 → (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))) ↔ ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
10	9	ralbidv 3190	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))) ↔ ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
11	10	riotabidv 7379	. . 3 ⊢ (𝑥 = 𝑋 → (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
12	1, 11	csbiegf 3887	. 2 ⊢ (𝑋 ∈ 𝐵 → ⦋𝑋 / 𝑥⦌(℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
13		cdleme31so.o	. . 3 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
14	13	csbeq2i 3862	. 2 ⊢ ⦋𝑋 / 𝑥⦌𝑂 = ⦋𝑋 / 𝑥⦌(℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
15		cdleme31so.c	. 2 ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))
16	12, 14, 15	3eqtr4g 2825	1 ⊢ (𝑋 ∈ 𝐵 → ⦋𝑋 / 𝑥⦌𝑂 = 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ⦋csb 3854   class class class wbr 5111  ℩crio 7376  (class class class)co 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6497  df-fv 6549  df-riota 7377  df-ov 7423

This theorem is used by:  cdleme31fv1s  41229  cdlemefrs32fva  41237  cdleme32fva  41274