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Mirrors > Home > ILE Home > Th. List > sbex | GIF version |
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Ref | Expression |
---|---|
sbex | ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbexyz 1928 | . . . 4 ⊢ ([𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑤 / 𝑦]𝜑) | |
2 | 1 | sbbii 1696 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑) |
3 | sbexyz 1928 | . . 3 ⊢ ([𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑) |
5 | ax-17 1465 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑤∃𝑥𝜑) | |
6 | 5 | sbco2v 1870 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦]∃𝑥𝜑) |
7 | ax-17 1465 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
8 | 7 | sbco2v 1870 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) |
9 | 8 | exbii 1542 | . 2 ⊢ (∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
10 | 4, 6, 9 | 3bitr3i 209 | 1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1427 [wsb 1693 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 |
This theorem is referenced by: sbabel 2255 sbcex2 2893 sbcexg 2894 |
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