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Mirrors > Home > ILE Home > Th. List > sbalv | GIF version |
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
sbalv.1 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbalv | ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbal 1931 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑) | |
2 | sbalv.1 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | |
3 | 2 | albii 1411 | . 2 ⊢ (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓) |
4 | 1, 3 | bitri 183 | 1 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1294 [wsb 1699 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 |
This theorem is referenced by: sbmo 2014 sbabel 2261 peano2 4438 |
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