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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfe1 | Structured version Visualization version GIF version |
Description: See nfe1 2153. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnfe1 | ⊢ Ⅎ'𝑥∃𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-modal4e 34070 | . 2 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) | |
2 | hbe1 2146 | . 2 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
3 | df-bj-nnf 34077 | . 2 ⊢ (Ⅎ'𝑥∃𝑥𝜑 ↔ ((∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) ∧ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑))) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ Ⅎ'𝑥∃𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∃wex 1779 Ⅎ'wnnf 34076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-bj-nnf 34077 |
This theorem is referenced by: (None) |
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