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Mirrors > Home > MPE Home > Th. List > hbsb3 | Structured version Visualization version GIF version |
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 14-May-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 2017 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑) |
3 | hbsb2a 2436 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | syl 17 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1599 [wsb 2011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-10 2134 ax-12 2162 ax-13 2333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-sb 2012 |
This theorem is referenced by: nfs1 2440 axc16ALT 2441 |
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