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Mirrors > Home > MPE Home > Th. List > spimefv | Structured version Visualization version GIF version |
Description: Version of spime 2387 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
spimefv.1 | ⊢ Ⅎ𝑥𝜑 |
spimefv.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimefv | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spimefv.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
3 | spimefv.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 2, 3 | spimedv 2189 | . 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓)) |
5 | 4 | mptru 1547 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊤wtru 1541 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-tru 1543 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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