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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme31se2 | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.) |
| Ref | Expression |
|---|---|
| cdleme31se2.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) |
| cdleme31se2.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
| Ref | Expression |
|---|---|
| cdleme31se2 | ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = 𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑡(𝑃 ∨ 𝑄) | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑡 ∧ | |
| 3 | nfcsb1v 3923 | . . . . . 6 ⊢ Ⅎ𝑡⦋𝑆 / 𝑡⦌𝐷 | |
| 4 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑡 ∨ | |
| 5 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑡((𝑅 ∨ 𝑆) ∧ 𝑊) | |
| 6 | 3, 4, 5 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑡(⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
| 7 | 1, 2, 6 | nfov 7461 | . . . 4 ⊢ Ⅎ𝑡((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝐴 → Ⅎ𝑡((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))) |
| 9 | csbeq1a 3913 | . . . . 5 ⊢ (𝑡 = 𝑆 → 𝐷 = ⦋𝑆 / 𝑡⦌𝐷) | |
| 10 | oveq2 7439 | . . . . . 6 ⊢ (𝑡 = 𝑆 → (𝑅 ∨ 𝑡) = (𝑅 ∨ 𝑆)) | |
| 11 | 10 | oveq1d 7446 | . . . . 5 ⊢ (𝑡 = 𝑆 → ((𝑅 ∨ 𝑡) ∧ 𝑊) = ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
| 12 | 9, 11 | oveq12d 7449 | . . . 4 ⊢ (𝑡 = 𝑆 → (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)) = (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
| 13 | 12 | oveq2d 7447 | . . 3 ⊢ (𝑡 = 𝑆 → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))) |
| 14 | 8, 13 | csbiegf 3932 | . 2 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))) |
| 15 | cdleme31se2.e | . . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) | |
| 16 | 15 | csbeq2i 3907 | . 2 ⊢ ⦋𝑆 / 𝑡⦌𝐸 = ⦋𝑆 / 𝑡⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) |
| 17 | cdleme31se2.y | . 2 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) | |
| 18 | 14, 16, 17 | 3eqtr4g 2802 | 1 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ⦋csb 3899 (class class class)co 7431 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: cdlemeg47rv2 40512 |
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