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Mirrors > Home > MPE Home > Th. List > spimv | Structured version Visualization version GIF version |
Description: A version of spim 2254 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2115 and spimvw 1927. See also spimvALT 2258. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2019. (Revised by BJ, 29-Nov-2020.) |
Ref | Expression |
---|---|
spimv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2250 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | spimv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | eximii 1764 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
4 | 3 | 19.36iv 1905 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: spv 2260 aevALTOLD 2321 axc16i 2322 reu6 3395 el 4847 aev-o 34216 axc11next 38607 |
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