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Mirrors > Home > MPE Home > Th. List > spimv | Structured version Visualization version GIF version |
Description: A version of spim 2242 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2101 and spimvw 1914. See also spimvALT 2246. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2006. (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 2238 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | spimv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | eximii 1754 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
4 | 3 | 19.36iv 1892 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: spv 2248 aevALTOLD 2309 axc16i 2310 reu6 3362 el 4773 aev-o 33234 axc11next 37629 |
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