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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme31se | Structured version Visualization version GIF version |
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.) |
Ref | Expression |
---|---|
cdleme31se.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊))) |
cdleme31se.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊))) |
Ref | Expression |
---|---|
cdleme31se | ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐸 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvd 2752 | . . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊)))) | |
2 | oveq1 6556 | . . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑠 ∨ 𝑇) = (𝑅 ∨ 𝑇)) | |
3 | 2 | oveq1d 6564 | . . . . 5 ⊢ (𝑠 = 𝑅 → ((𝑠 ∨ 𝑇) ∧ 𝑊) = ((𝑅 ∨ 𝑇) ∧ 𝑊)) |
4 | 3 | oveq2d 6565 | . . . 4 ⊢ (𝑠 = 𝑅 → (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊)) = (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊))) |
5 | 4 | oveq2d 6565 | . . 3 ⊢ (𝑠 = 𝑅 → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊)))) |
6 | 1, 5 | csbiegf 3523 | . 2 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊)))) |
7 | cdleme31se.e | . . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊))) | |
8 | 7 | csbeq2i 3945 | . 2 ⊢ ⦋𝑅 / 𝑠⦌𝐸 = ⦋𝑅 / 𝑠⦌((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊))) |
9 | cdleme31se.y | . 2 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊))) | |
10 | 6, 8, 9 | 3eqtr4g 2669 | 1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐸 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⦋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 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 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: cdleme31sde 34691 cdleme31sn1c 34694 |
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