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Mirrors > Home > MPE Home > Th. List > Mathboxes > or2expropbilem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for or2expropbi 46984 and ich2exprop 47396. (Contributed by AV, 16-Jul-2023.) |
Ref | Expression |
---|---|
or2expropbilem2 | ⊢ (∃𝑎∃𝑏(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1912 | . 2 ⊢ Ⅎ𝑥(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) | |
2 | nfv 1912 | . 2 ⊢ Ⅎ𝑦(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) | |
3 | nfv 1912 | . . 3 ⊢ Ⅎ𝑎〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 | |
4 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑎𝑦 | |
5 | nfsbc1v 3811 | . . . 4 ⊢ Ⅎ𝑎[𝑥 / 𝑎]𝜑 | |
6 | 4, 5 | nfsbcw 3813 | . . 3 ⊢ Ⅎ𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑 |
7 | 3, 6 | nfan 1897 | . 2 ⊢ Ⅎ𝑎(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) |
8 | nfv 1912 | . . 3 ⊢ Ⅎ𝑏〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 | |
9 | nfsbc1v 3811 | . . 3 ⊢ Ⅎ𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑 | |
10 | 8, 9 | nfan 1897 | . 2 ⊢ Ⅎ𝑏(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) |
11 | opeq12 4880 | . . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉) | |
12 | 11 | eqeq2d 2746 | . . 3 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ↔ 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉)) |
13 | sbceq1a 3802 | . . . 4 ⊢ (𝑎 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑎]𝜑)) | |
14 | sbceq1a 3802 | . . . 4 ⊢ (𝑏 = 𝑦 → ([𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)) | |
15 | 13, 14 | sylan9bb 509 | . . 3 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)) |
16 | 12, 15 | anbi12d 632 | . 2 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) ↔ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))) |
17 | 1, 2, 7, 10, 16 | cbvex2v 2345 | 1 ⊢ (∃𝑎∃𝑏(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 [wsbc 3791 〈cop 4637 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: or2expropbi 46984 ich2exprop 47396 |
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