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Mirrors > Home > ILE Home > Th. List > hbsb3 | GIF version |
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbsb3.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
hbsb3 | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsb3.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | sbimi 1647 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑) |
3 | hbsb2a 1687 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | syl 14 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 [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-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-11 1397 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: nfs1 1690 sbcof2 1691 ax16 1694 sb8h 1734 sb8eh 1735 ax16ALT 1739 |
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