Theorem cdleme31sdnN 40369

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 261	. . 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 40366	. . . 4 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = 𝐶)
6	5	elv 3482	. . 3 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = 𝐶
7	2, 6	ifbieq2i 4555	. 2 ⊢ if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
8	1, 7	eqtr4i 2765	1 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷)

Syntax hints:   = wceq 1536  Vcvv 3477  ⦋csb 3907  ifcif 4530   class class class wbr 5147  (class class class)co 7430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433

This theorem is referenced by: (None)