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Mirrors > Home > MPE Home > Th. List > sb4b | Structured version Visualization version GIF version |
Description: Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Ref | Expression |
---|---|
sb4b | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4 2503 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | sb2 2498 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | impbid1 215 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1629 [wsb 2049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-ex 1853 df-nf 1858 df-sb 2050 |
This theorem is referenced by: sbcom3 2558 sbal1 2608 sbal2 2609 wl-sb6nae 33674 wl-sbalnae 33679 |
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