Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2024 | . 2 ⊢ Ⅎ𝑥𝜑 |
3 | 2 | sbf 2380 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 [wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |