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Mirrors > Home > NFE Home > Th. List > speiv | GIF version |
Description: Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.) |
Ref | Expression |
---|---|
speiv.1 | ⊢ (x = y → (φ ↔ ψ)) |
speiv.2 | ⊢ ψ |
Ref | Expression |
---|---|
speiv | ⊢ ∃xφ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | speiv.2 | . 2 ⊢ ψ | |
2 | speiv.1 | . . . 4 ⊢ (x = y → (φ ↔ ψ)) | |
3 | 2 | biimprd 214 | . . 3 ⊢ (x = y → (ψ → φ)) |
4 | 3 | spimev 1999 | . 2 ⊢ (ψ → ∃xφ) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ∃xφ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∃wex 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: ce0addcnnul 6179 |
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