Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbv | Structured version Visualization version GIF version |
Description: Substitution for a variable not occurring in a proposition. See sbf 2270 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2097 can be proved directly by chaining equsv 2008 with sb6 2092. (Contributed by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
sbv | ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbe 2087 | . . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) | |
2 | ax5e 1912 | . . 3 ⊢ (∃𝑥𝜑 → 𝜑) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑) |
4 | ax-5 1910 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
5 | stdpc4 2072 | . . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → [𝑡 / 𝑥]𝜑) |
7 | 3, 6 | impbii 211 | 1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1534 ∃wex 1779 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 |
This theorem depends on definitions: df-bi 209 df-ex 1780 df-sb 2069 |
This theorem is referenced by: sbcom4 2098 sbievw2 2106 iuninc 30312 measiuns 31497 ballotlemodife 31776 bj-vjust 34370 mptsnunlem 34643 ichv 43683 ichf 43684 |
Copyright terms: Public domain | W3C validator |