Theorem cdleme31sn 39554

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 1915	. . . . 5 ⊢ Ⅎ𝑠 𝑅 ≤ (𝑃 ∨ 𝑄)
2		nfcsb1v 3917	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐼
3		nfcsb1v 3917	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐷
4	1, 2, 3	nfif 4557	. . . 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 5150	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄)))
7		csbeq1a 3906	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐼 = ⦋𝑅 / 𝑠⦌𝐼)
8		csbeq1a 3906	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐷 = ⦋𝑅 / 𝑠⦌𝐷)
9	6, 7, 8	ifbieq12d 4555	. . 3 ⊢ (𝑠 = 𝑅 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
10	5, 9	csbiegf 3926	. 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 3900	. 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 2795	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2881  ⦋csb 3892  ifcif 4527   class class class wbr 5147  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148

This theorem is referenced by:  cdleme31sn1  39555  cdleme31sn2  39563