Theorem cdleme31sn 38321

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 1918	. . . . 5 ⊢ Ⅎ𝑠 𝑅 ≤ (𝑃 ∨ 𝑄)
2		nfcsb1v 3853	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐼
3		nfcsb1v 3853	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐷
4	1, 2, 3	nfif 4486	. . . 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 5073	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄)))
7		csbeq1a 3842	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐼 = ⦋𝑅 / 𝑠⦌𝐼)
8		csbeq1a 3842	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐷 = ⦋𝑅 / 𝑠⦌𝐷)
9	6, 7, 8	ifbieq12d 4484	. . 3 ⊢ (𝑠 = 𝑅 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
10	5, 9	csbiegf 3862	. 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 3836	. 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 2804	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  ⦋csb 3828  ifcif 4456   class class class wbr 5070  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071

This theorem is referenced by:  cdleme31sn1  38322  cdleme31sn2  38330