Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sb4OLD | Structured version Visualization version GIF version |
Description: Obsolete as of 30-Jul-2023. Use sb4b 2499 instead. One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.) Revise df-sb 2070. (Revised by Wolf Lammen, 25-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb4OLD | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4b 2499 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | 1 | biimpd 231 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 [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 ax-13 2390 |
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: sbequiOLD 2534 sbi1OLD 2542 |
Copyright terms: Public domain | W3C validator |