Theorem cdleme31sn 41628

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)

Hypotheses

Ref	Expression
cdleme31sn.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme31sn.c	⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)

Assertion

Ref	Expression
cdleme31sn	⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ≤ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠

Allowed substitution hints:   𝐶(𝑠)   𝐷(𝑠)   𝐼(𝑠)   𝑁(𝑠)

Proof of Theorem cdleme31sn

Step	Hyp	Ref	Expression
1		nfv 1949	. . . . 5 ⊢ Ⅎ𝑠 𝑅 ≤ (𝑃 ∨ 𝑄)
2		nfcsb1v 3875	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐼
3		nfcsb1v 3875	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐷
4	1, 2, 3	nfif 4516	. . . 4 ⊢ Ⅎ𝑠if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)
5	4	a1i 11	. . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
6		breq1 5109	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄)))
7		csbeq1a 3865	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐼 = ⦋𝑅 / 𝑠⦌𝐼)
8		csbeq1a 3865	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐷 = ⦋𝑅 / 𝑠⦌𝐷)
9	6, 7, 8	ifbieq12d 4514	. . 3 ⊢ (𝑠 = 𝑅 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
10	5, 9	csbiegf 3884	. 2 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
11		cdleme31sn.n	. . 3 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
12	11	csbeq2i 3859	. 2 ⊢ ⦋𝑅 / 𝑠⦌𝑁 = ⦋𝑅 / 𝑠⦌if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
13		cdleme31sn.c	. 2 ⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)
14	10, 12, 13	3eqtr4g 2823	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  ⦋csb 3851  ifcif 4485   class class class wbr 5106  (class class class)co 7440

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107

This theorem is used by:  cdleme31sn1  41629  cdleme31sn2  41637