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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 1974 and spimevw 1997 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 2370. Use spimefv 2190 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 2386 | . 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 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-nf 1785 |
This theorem is referenced by: spimev 2390 exnel 35244 |
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