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Mirrors > Home > MPE Home > Th. List > sbh | Structured version Visualization version GIF version |
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.) |
Ref | Expression |
---|---|
sbh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
sbh | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nf5i 2141 | . 2 ⊢ Ⅎ𝑥𝜑 |
3 | 2 | sbf 2261 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: (None) |
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