Theorem cdleme31sdnN 41189

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme31sdn.c	⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme31sdn.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme31sdn.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)

Assertion

Ref	Expression
cdleme31sdnN	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷)

Distinct variable groups:   𝑡, ∨   𝑡, ∧   𝑡,𝑃   𝑡,𝑄   𝑡,𝑈   𝑡,𝑊   𝑡,𝑠

Allowed substitution hints:   𝐶(𝑡, 𝑠)   𝐷(𝑡, 𝑠)   𝑃(𝑠)   𝑄(𝑠)   𝑈(𝑠)   𝐼(𝑡, 𝑠)   ∨ (𝑠)   ≤ (𝑡, 𝑠)   ∧ (𝑠)   𝑁(𝑡, 𝑠)   𝑊(𝑠)

Proof of Theorem cdleme31sdnN

Step	Hyp	Ref	Expression
1		cdleme31sdn.n	. 2 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
2		biid 264	. . 3 ⊢ (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑠 ≤ (𝑃 ∨ 𝑄))
3		cdleme31sdn.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
4		cdleme31sdn.c	. . . . 5 ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
5	3, 4	cdleme31sc 41186	. . . 4 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = 𝐶)
6	5	elv 3459	. . 3 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = 𝐶
7	2, 6	ifbieq2i 4512	. 2 ⊢ if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
8	1, 7	eqtr4i 2788	1 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷)

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3454  ⦋csb 3852  ifcif 4486   class class class wbr 5108  (class class class)co 7412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415

This theorem is used by: (None)