Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > expandexn | Structured version Visualization version GIF version |
Description: Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
expandexn.1 | ⊢ (𝜑 ↔ ¬ 𝜓) |
Ref | Expression |
---|---|
expandexn | ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expandexn.1 | . . 3 ⊢ (𝜑 ↔ ¬ 𝜓) | |
2 | 1 | exbii 1853 | . 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥 ¬ 𝜓) |
3 | exnal 1832 | . 2 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1539 ∃wex 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 |
This theorem depends on definitions: df-bi 206 df-ex 1786 |
This theorem is referenced by: rr-grothprimbi 41866 |
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