![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbft | Structured version Visualization version GIF version |
Description: Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) |
Ref | Expression |
---|---|
sbft | ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbe 2086 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) | |
2 | 19.9t 2198 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | imbitrid 243 | . 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝜑)) |
4 | nf5r 2188 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
5 | stdpc4 2072 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
6 | 4, 5 | syl6 35 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
7 | 3, 6 | impbid 211 | 1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 df-sb 2069 |
This theorem is referenced by: sbf 2263 sbctt 3820 wl-sbrimt 36034 wl-sblimt 36035 wl-equsb4 36041 ichnfimlem 45729 |
Copyright terms: Public domain | W3C validator |