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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.57 | Structured version Visualization version GIF version |
Description: Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm11.57 | ⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfal 2331 | . . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑 |
3 | sp 2180 | . . . . 5 ⊢ (∀𝑥𝜑 → 𝜑) | |
4 | stdpc4 2073 | . . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
5 | 3, 4 | jca 515 | . . . 4 ⊢ (∀𝑥𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
6 | 2, 5 | alrimi 2211 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
7 | 6 | axc4i 2330 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
8 | simpl 486 | . . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑) | |
9 | 8 | sps 2182 | . . 3 ⊢ (∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑) |
10 | 9 | alimi 1813 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → ∀𝑥𝜑) |
11 | 7, 10 | impbii 212 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1536 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 |
This theorem is referenced by: (None) |
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