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 2228. See 19.36iv 1955 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 2228 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | mpbi 233 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-nf 1792 |
This theorem is referenced by: spimfv 2237 spim 2386 vtoclf 3470 bj-vtoclf 34834 |
Copyright terms: Public domain | W3C validator |