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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 2335 among others. For this and the following theorems, see also 19.35 1874, 19.21 2203, 19.23 2207. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.) |
Ref | Expression |
---|---|
bj-biexal1 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜓 | |
2 | 1 | 19.23 2207 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1777 df-nf 1781 |
This theorem is referenced by: bj-biexal3 34036 |
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