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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon2lem1 | Structured version Visualization version GIF version |
Description: Lemma for dfon2 30941. (Contributed by Scott Fenton, 28-Feb-2011.) |
Ref | Expression |
---|---|
dfon2lem1 | ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | truni 4695 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}) | |
2 | nfsbc1v 3422 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑥Tr 𝑦 | |
4 | nfsbc1v 3422 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓 | |
5 | 2, 3, 4 | nf3an 1819 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓) |
6 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | sbceq1a 3413 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | treq 4686 | . . . . 5 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
9 | sbceq1a 3413 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓)) | |
10 | 7, 8, 9 | 3anbi123d 1391 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓))) |
11 | 5, 6, 10 | elabf 3318 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)) |
12 | 11 | simp2bi 1070 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦) |
13 | 1, 12 | mprg 2910 | 1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1031 ∈ wcel 1977 {cab 2596 [wsbc 3402 ∪ cuni 4372 Tr wtr 4680 |
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-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-in 3547 df-ss 3554 df-uni 4373 df-iun 4457 df-tr 4681 |
This theorem is referenced by: dfon2lem3 30934 dfon2lem7 30938 |
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