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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-biexex | Structured version Visualization version GIF version |
Description: A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.) |
Ref | Expression |
---|---|
bj-biexex | ⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2088 | . 2 ⊢ Ⅎ𝑥∃𝑥𝜓 | |
2 | 1 | 19.23 2142 | 1 ⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1506 ∃wex 1743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-10 2080 ax-12 2107 |
This theorem depends on definitions: df-bi 199 df-ex 1744 df-nf 1748 |
This theorem is referenced by: (None) |
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