| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > pm11.53 | Structured version Visualization version GIF version | ||
| Description: Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1944 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.) |
| Ref | Expression |
|---|---|
| pm11.53 | ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.21v 1939 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) | |
| 2 | 1 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)) |
| 3 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 4 | 3 | nfal 2323 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
| 5 | 4 | 19.23 2211 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
| 6 | 2, 5 | bitri 275 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |