Theorem cdleme31sn1c 39259

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.)

Hypotheses

Ref	Expression
cdleme31sn1c.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme31sn1c.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme31sn1c.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme31sn1c.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
cdleme31sn1c.c	⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))

Assertion

Ref	Expression
cdleme31sn1c	⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Distinct variable groups:   𝑡,𝑠,𝑦,𝐴   𝐵,𝑠   𝐸,𝑠   ∨ ,𝑠,𝑡,𝑦   ≤ ,𝑠,𝑡,𝑦   ∧ ,𝑠   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑊,𝑠

Allowed substitution hints:   𝐵(𝑦,𝑡)   𝐶(𝑦,𝑡,𝑠)   𝐷(𝑦,𝑡,𝑠)   𝐸(𝑦,𝑡)   𝐺(𝑦,𝑡,𝑠)   𝐼(𝑦,𝑡,𝑠)   ∧ (𝑦,𝑡)   𝑁(𝑦,𝑡,𝑠)   𝑊(𝑦,𝑡)   𝑌(𝑦,𝑡,𝑠)

Proof of Theorem cdleme31sn1c

Step	Hyp	Ref	Expression
1		cdleme31sn1c.i	. . 3 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
2		cdleme31sn1c.n	. . 3 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
3		eqid 2733	. . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))
4	1, 2, 3	cdleme31sn1 39252	. 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
5		cdleme31sn1c.g	. . . . . . . . 9 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
6		cdleme31sn1c.y	. . . . . . . . 9 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
7	5, 6	cdleme31se 39253	. . . . . . . 8 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐺 = 𝑌)
8	7	adantr 482	. . . . . . 7 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝐺 = 𝑌)
9	8	eqeq2d 2744	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑦 = ⦋𝑅 / 𝑠⦌𝐺 ↔ 𝑦 = 𝑌))
10	9	imbi2d 341	. . . . 5 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺) ↔ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)))
11	10	ralbidv 3178	. . . 4 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺) ↔ ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)))
12	11	riotabidv 7367	. . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)))
13		cdleme31sn1c.c	. . 3 ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))
14	12, 13	eqtr4di 2791	. 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) = 𝐶)
15	4, 14	eqtrd 2773	1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ⦋csb 3894  ifcif 4529   class class class wbr 5149  ℩crio 7364  (class class class)co 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-riota 7365  df-ov 7412

This theorem is referenced by:  cdlemefs32sn1aw  39285  cdleme43fsv1snlem  39291  cdleme41sn3a  39304  cdleme40m  39338  cdleme40n  39339