![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > spimv | Structured version Visualization version GIF version |
Description: A version of spim 2390 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2237 and spimvw 1993 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2375. Use spimvw 1993 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
spimv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1912 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | spimv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | spim 2390 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 |
This theorem is referenced by: spv 2396 |
Copyright terms: Public domain | W3C validator |