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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 | csbnestg 3950 | . 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 34690 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌𝑁 = 𝑂) |
5 | 4 | csbeq1d 3506 | . . 3 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = ⦋𝑂 / 𝑡⦌𝐷) |
6 | ovex 6577 | . . . . 5 ⊢ ((𝑆 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊))) ∈ V | |
7 | 3, 6 | eqeltri 2684 | . . . 4 ⊢ 𝑂 ∈ V |
8 | cdleme31snd.d | . . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
9 | cdleme31snd.e | . . . . 5 ⊢ 𝐸 = ((𝑂 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑂) ∧ 𝑊))) | |
10 | 8, 9 | cdleme31sc 34690 | . . . 4 ⊢ (𝑂 ∈ V → ⦋𝑂 / 𝑡⦌𝐷 = 𝐸) |
11 | 7, 10 | ax-mp 5 | . . 3 ⊢ ⦋𝑂 / 𝑡⦌𝐷 = 𝐸 |
12 | 5, 11 | syl6eq 2660 | . 2 ⊢ (𝑆 ∈ 𝐴 → ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷 = 𝐸) |
13 | 1, 12 | eqtrd 2644 | 1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: cdlemeg46ngfr 34824 |
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