![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sbalexi | Structured version Visualization version GIF version |
Description: Inference form of sbalex 2238, avoiding ax-10 2136 by using ax-gen 1793. (Contributed by SN, 12-Aug-2025.) |
Ref | Expression |
---|---|
sbalexi.1 | ⊢ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) |
Ref | Expression |
---|---|
sbalexi | ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbalexi.1 | . . 3 ⊢ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) | |
2 | ax12ev2 2176 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝑥 = 𝑦 → 𝜑) |
4 | 3 | ax-gen 1793 | 1 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃wex 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-12 2173 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1778 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |