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| Mirrors > Home > MPE Home > Th. List > Mathboxes > or2expropbilem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for or2expropbi 47046 and ich2exprop 47458. (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 1914 | . 2 ⊢ Ⅎ𝑥(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) | |
| 2 | nfv 1914 | . 2 ⊢ Ⅎ𝑦(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) | |
| 3 | nfv 1914 | . . 3 ⊢ Ⅎ𝑎〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 | |
| 4 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑎𝑦 | |
| 5 | nfsbc1v 3808 | . . . 4 ⊢ Ⅎ𝑎[𝑥 / 𝑎]𝜑 | |
| 6 | 4, 5 | nfsbcw 3810 | . . 3 ⊢ Ⅎ𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑 | 
| 7 | 3, 6 | nfan 1899 | . 2 ⊢ Ⅎ𝑎(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) | 
| 8 | nfv 1914 | . . 3 ⊢ Ⅎ𝑏〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 | |
| 9 | nfsbc1v 3808 | . . 3 ⊢ Ⅎ𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑 | |
| 10 | 8, 9 | nfan 1899 | . 2 ⊢ Ⅎ𝑏(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) | 
| 11 | opeq12 4875 | . . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉) | |
| 12 | 11 | eqeq2d 2748 | . . 3 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ↔ 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉)) | 
| 13 | sbceq1a 3799 | . . . 4 ⊢ (𝑎 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑎]𝜑)) | |
| 14 | sbceq1a 3799 | . . . 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 2346 | 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 1540 ∃wex 1779 [wsbc 3788 〈cop 4632 | 
| 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-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 | 
| This theorem is referenced by: or2expropbi 47046 ich2exprop 47458 | 
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