![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > spimevw | Structured version Visualization version GIF version |
Description: Existential introduction, using implicit substitution. This is to spimew 1976 what spimvw 2000 is to spimw 1975. Version of spimev 2391 and spimefv 2192 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 1914 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | spimevw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | spimew 1976 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1782 |
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 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: dtruALT2 5330 zfpair 5381 exneq 5397 dtruOLD 5403 fvn0ssdmfun 7030 onsupmaxb 41602 refimssco 41953 rlimdmafv 45483 rlimdmafv2 45564 elsprel 45741 |
Copyright terms: Public domain | W3C validator |