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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax12v3 | Structured version Visualization version GIF version |
Description: A weak version of ax-12 2170 which is stronger than ax12v 2171. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2014), then bj-ax12v3 35867 implies ax-5 1912 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 35868. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ax12v3 | ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1912 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | ax12 2421 | . 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 1538 |
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 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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