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Mirrors > Home > MPE Home > Th. List > spimv | Structured version Visualization version GIF version |
Description: A version of spim 2378 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2224 and spimvw 1991 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2363. Use spimvw 1991 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
spimv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | spimv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | spim 2378 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 |
This theorem is referenced by: spv 2384 |
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