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Mirrors > Home > MPE Home > Th. List > spimevw | Structured version Visualization version GIF version |
Description: Existential introduction, using implicit substitution. This is to spimew 1974 what spimvw 2002 is to spimw 1973. Version of spimev 2410 and spimefv 2198 with an additional disjoint variable condition, using only Tarski's FOL axiom schemes. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 17-Mar-2020.) |
Ref | Expression |
---|---|
spimevw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimevw | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1911 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | spimevw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | spimew 1974 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1780 |
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 1911 ax-6 1970 |
This theorem depends on definitions: df-bi 209 df-ex 1781 |
This theorem is referenced by: zfpair 5322 fvn0ssdmfun 6842 bj-dtru 34139 sn-dtru 39160 refimssco 40016 rlimdmafv 43425 rlimdmafv2 43506 elsprel 43686 |
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