Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbnvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbn 2287 as of 8-Jul-2023. Substitution is not affected by negation. Version of sbn 2287 with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Wolf Lammen, 18-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbnvOLD | ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exanali 1859 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sb5 2276 | . 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) | |
3 | sb6 2093 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
4 | 3 | notbii 322 | . 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 1, 2, 4 | 3bitr4i 305 | 1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: sbi2vOLD 2324 |
Copyright terms: Public domain | W3C validator |