Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 19.36i | Structured version Visualization version GIF version |
Description: Inference associated with 19.36 2223. See 19.36iv 1950 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.) |
Ref | Expression |
---|---|
19.36.1 | ⊢ Ⅎ𝑥𝜓 |
19.36i.2 | ⊢ ∃𝑥(𝜑 → 𝜓) |
Ref | Expression |
---|---|
19.36i | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.36i.2 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) | |
2 | 19.36.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.36 2223 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | mpbi 229 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 Ⅎwnf 1786 |
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 1913 ax-6 1971 ax-7 2011 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 |
This theorem is referenced by: spimfv 2232 spim 2387 vtoclf 3495 bj-vtoclf 35086 |
Copyright terms: Public domain | W3C validator |