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| Mirrors > Home > MPE Home > Th. List > spimv | Structured version Visualization version GIF version | ||
| Description: A version of spim 2392 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2239 and spimvw 1995 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use spimvw 1995 instead. (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| spimv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
| Ref | Expression |
|---|---|
| spimv | ⊢ (∀𝑥𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 2 | spimv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
| 3 | 1, 2 | spim 2392 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: spv 2398 |
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