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Mirrors > Home > ILE Home > Th. List > sbal | GIF version |
Description: Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Ref | Expression |
---|---|
sbal | ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbalyz 1972 | . . . 4 ⊢ ([𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑤 / 𝑦]𝜑) | |
2 | 1 | sbbii 1738 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑) |
3 | sbalyz 1972 | . . 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 1506 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑤∀𝑥𝜑) | |
6 | 5 | sbco2vh 1916 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑) |
7 | ax-17 1506 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
8 | 7 | sbco2vh 1916 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) |
9 | 8 | albii 1446 | . 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 ∀wal 1329 [wsb 1735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: sbal1 1975 sbalv 1978 sbcal 2955 sbcalg 2956 |
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