Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.9htbi | Structured version Visualization version GIF version |
Description: Strengthening 19.9ht 2330 by replacing its succedent with a biconditional (19.9t 2194 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.) |
Ref | Expression |
---|---|
bj-19.9htbi | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9ht 2330 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | |
2 | 19.8a 2170 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
3 | 1, 2 | impbid1 226 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: bj-hbntbi 33935 |
Copyright terms: Public domain | W3C validator |