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Mirrors > Home > MPE Home > Th. List > spime | Structured version Visualization version GIF version |
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1967 and spimevw 1990 for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Usage of this theorem is discouraged because it depends on ax-13 2363. Use spimefv 2183 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
spime.1 | ⊢ Ⅎ𝑥𝜑 |
spime.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spime | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spime.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
3 | spime.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 2, 3 | spimed 2379 | . 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓)) |
5 | 4 | mptru 1540 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊤wtru 1534 ∃wex 1773 Ⅎwnf 1777 |
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-tru 1536 df-ex 1774 df-nf 1778 |
This theorem is referenced by: spimev 2383 exnel 35270 |
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