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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2stdpc5 | Structured version Visualization version GIF version |
Description: A double stdpc5 2198 (one direction of PM*11.3). See also 2stdpc4 2066 and 19.21vv 40585. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
2stdpc5.1 | ⊢ Ⅎ𝑥𝜑 |
2stdpc5.2 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
2stdpc5 | ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2stdpc5.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | stdpc5 2198 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑦𝜓)) |
3 | 2 | alimi 1803 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓)) |
4 | 2stdpc5.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | stdpc5 2198 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓)) |
6 | 3, 5 | syl 17 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (𝜑 → ∀𝑥∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: ax11-pm 34052 ax11-pm2 34056 |
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