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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.59 | Structured version Visualization version GIF version |
Description: Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
pm11.59 | ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1995 | . . 3 ⊢ Ⅎ𝑦(𝜑 → 𝜓) | |
2 | 1 | nfal 2317 | . 2 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝜓) |
3 | sp 2207 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
4 | spsbim 2541 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
5 | 3, 4 | anim12d 596 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) |
6 | 5 | axc4i 2295 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) |
7 | 2, 6 | alrimi 2238 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∀wal 1629 [wsb 2049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-ex 1853 df-nf 1858 df-sb 2050 |
This theorem is referenced by: (None) |
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