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Mirrors > Home > MPE Home > Th. List > spimed | Structured version Visualization version GIF version |
Description: Deduction version of spime 2388. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2371. Use spimedv 2191 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
spimed.1 | ⊢ (𝜒 → Ⅎ𝑥𝜑) |
spimed.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimed | ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spimed.1 | . . 3 ⊢ (𝜒 → Ⅎ𝑥𝜑) | |
2 | 1 | nf5rd 2190 | . 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑)) |
3 | ax6e 2382 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
4 | spimed.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
5 | 3, 4 | eximii 1840 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
6 | 5 | 19.35i 1882 | . 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓) |
7 | 2, 6 | syl6 35 | 1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: spime 2388 2ax6elem 2469 |
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