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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 2011 | . . . 4 ⊢ ([𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑤 / 𝑦]𝜑) | |
2 | 1 | sbbii 1776 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑) |
3 | sbalyz 2011 | . . 3 ⊢ ([𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑) | |
4 | 2, 3 | bitri 184 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑) |
5 | ax-17 1537 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑤∀𝑥𝜑) | |
6 | 5 | sbco2vh 1957 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑) |
7 | ax-17 1537 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
8 | 7 | sbco2vh 1957 | . . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) |
9 | 8 | albii 1481 | . 2 ⊢ (∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
10 | 4, 6, 9 | 3bitr3i 210 | 1 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∀wal 1362 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: sbal1 2014 sbalv 2017 sbcal 3029 sbcalg 3030 |
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