Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax12v3 | Structured version Visualization version GIF version |
Description: A weak version of ax-12 2167 which is stronger than ax12v 2168. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2010), then bj-ax12v3 33916 implies ax-5 1902 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 33917. (Contributed by BJ, 6-Jul-2021.) |
Ref | Expression |
---|---|
bj-ax12v3 | ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1902 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | ax12 2437 | . 2 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 1, 2 | syl5 34 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 |
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 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |