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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 1952 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 1947 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) | |
2 | 1 | albii 1827 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)) |
3 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfal 2322 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
5 | 4 | 19.23 2209 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
6 | 2, 5 | bitri 278 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 ∃wex 1787 |
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-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-nf 1792 |
This theorem is referenced by: (None) |
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