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Mirrors > Home > MPE Home > Th. List > sbft | Structured version Visualization version GIF version |
Description: Substitution has no effect on a non-free 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 1884 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) | |
2 | 19.9t 2071 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | syl5ib 234 | . 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝜑)) |
4 | nf5r 2064 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
5 | stdpc4 2353 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
6 | 4, 5 | syl6 35 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
7 | 3, 6 | impbid 202 | 1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 [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-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: sbf 2380 sbctt 3500 wl-sbrimt 33331 wl-sblimt 33332 wl-equsb4 33338 |
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