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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. Usage of this theorem is discouraged because it depends on ax-13 2390. Theorem nfsb 2565 replaces the distinctor with a disjoint variable condition. Visit also nfsbv 2349 for a weaker version of nfsb 2565 not requiring ax-13 2390. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfsb4.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsb4 | ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsb4t 2539 | . 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) | |
2 | nfsb4.1 | . 2 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | mpg 1798 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 Ⅎwnf 1784 [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-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: sbco2 2553 nfsbOLD 2566 |
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