Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfsb4 | Structured version Visualization version GIF version |
Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
nfsb4.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsb4 | ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsb4t 2532 | . 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) | |
2 | nfsb4.1 | . 2 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | mpg 1789 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 Ⅎwnf 1775 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: sbco2 2546 nfsbOLD 2559 |
Copyright terms: Public domain | W3C validator |