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Mirrors > Home > MPE Home > Th. List > nfsbd | Structured version Visualization version GIF version |
Description: Deduction version of nfsb 2527. (Contributed by NM, 15-Feb-2013.) Usage of this theorem is discouraged because it depends on ax-13 2372. Use nfsbv 2324 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
nfsbd.1 | ⊢ Ⅎ𝑥𝜑 |
nfsbd.2 | ⊢ (𝜑 → Ⅎ𝑧𝜓) |
Ref | Expression |
---|---|
nfsbd | ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsbd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | nfsbd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
3 | 1, 2 | alrimi 2206 | . . 3 ⊢ (𝜑 → ∀𝑥Ⅎ𝑧𝜓) |
4 | nfsb4t 2503 | . . 3 ⊢ (∀𝑥Ⅎ𝑧𝜓 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)) |
6 | axc16nf 2255 | . 2 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) | |
7 | 5, 6 | pm2.61d2 181 | 1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 Ⅎwnf 1786 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 |
This theorem is referenced by: nfsb 2527 nfabd 2932 wl-sb8eut 35732 |
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