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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-biexal1 | Structured version Visualization version GIF version |
Description: A general FOL biconditional that generalizes 19.9ht 2128 among others. For this and the following theorems, see also 19.35 1794, 19.21 2062, 19.23 2067. (Contributed by BJ, 20-Oct-2019.) |
Ref | Expression |
---|---|
bj-biexal1 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2015 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜓 | |
2 | 1 | 19.23 2067 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-biexal3 31885 |
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