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Mirrors > Home > MPE Home > Th. List > 19.36i | Structured version Visualization version GIF version |
Description: Inference associated with 19.36 2222. See 19.36iv 1938 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.) |
Ref | Expression |
---|---|
19.36.1 | ⊢ Ⅎ𝑥𝜓 |
19.36i.2 | ⊢ ∃𝑥(𝜑 → 𝜓) |
Ref | Expression |
---|---|
19.36i | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.36i.2 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) | |
2 | 19.36.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.36 2222 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | mpbi 231 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∃wex 1771 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: spimfv 2231 spim 2396 vtoclf 3556 bj-vtoclf 34128 |
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