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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 1879 | . . . 4 ⊢ ([𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑤 / 𝑦]𝜑) | |
2 | 1 | sbbii 1648 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑) |
3 | sbexyz 1879 | . . 3 ⊢ ([𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑) | |
4 | 2, 3 | bitri 173 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑) |
5 | ax-17 1419 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑤∃𝑥𝜑) | |
6 | 5 | sbco2v 1821 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦]∃𝑥𝜑) |
7 | ax-17 1419 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
8 | 7 | sbco2v 1821 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) |
9 | 8 | exbii 1496 | . 2 ⊢ (∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
10 | 4, 6, 9 | 3bitr3i 199 | 1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1381 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: sbabel 2203 sbcex2 2812 sbcexg 2813 |
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