Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-biexal3 | Structured version Visualization version GIF version |
Description: When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.) |
Ref | Expression |
---|---|
bj-biexal3 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-biexal1 34039 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) | |
2 | bj-biexal2 34040 | . 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) | |
3 | 1, 2 | bitr4i 280 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |