Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme31snd | Structured version Visualization version GIF version |
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.) |
Ref | Expression |
---|---|
cdleme31snd.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
cdleme31snd.n | ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) |
cdleme31snd.e | ⊢ 𝐸 = ((𝑂 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑂) ∧ 𝑊))) |
cdleme31snd.o | ⊢ 𝑂 = ((𝑆 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊))) |
Ref | Expression |
---|---|
cdleme31snd | ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbnestgw 4373 | . 2 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷) | |
2 | cdleme31snd.n | . . . . 5 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) | |
3 | cdleme31snd.o | . . . . 5 ⊢ 𝑂 = ((𝑆 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊))) | |
4 | 2, 3 | cdleme31sc 37514 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌𝑁 = 𝑂) |
5 | 4 | csbeq1d 3887 | . . 3 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = ⦋𝑂 / 𝑡⦌𝐷) |
6 | 3 | ovexi 7184 | . . . 4 ⊢ 𝑂 ∈ V |
7 | cdleme31snd.d | . . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
8 | cdleme31snd.e | . . . . 5 ⊢ 𝐸 = ((𝑂 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑂) ∧ 𝑊))) | |
9 | 7, 8 | cdleme31sc 37514 | . . . 4 ⊢ (𝑂 ∈ V → ⦋𝑂 / 𝑡⦌𝐷 = 𝐸) |
10 | 6, 9 | ax-mp 5 | . . 3 ⊢ ⦋𝑂 / 𝑡⦌𝐷 = 𝐸 |
11 | 5, 10 | syl6eq 2872 | . 2 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = 𝐸) |
12 | 1, 11 | eqtrd 2856 | 1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⦋csb 3883 (class class class)co 7150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-ov 7153 |
This theorem is referenced by: cdlemeg46ngfr 37648 |
Copyright terms: Public domain | W3C validator |