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| 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 1913 | . . 3 ⊢ Ⅎ𝑦(𝜑 → 𝜓) | |
| 2 | 1 | nfal 2322 | . 2 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝜓) | 
| 3 | sp 2182 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
| 4 | spsbim 2071 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
| 5 | 3, 4 | anim12d 609 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) | 
| 6 | 5 | axc4i 2321 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) | 
| 7 | 2, 6 | alrimi 2212 | 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 395 ∀wal 1537 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: (None) | 
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