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Mirrors > Home > MPE Home > Th. List > ax12ev2 | Structured version Visualization version GIF version |
Description: Version of ax12v2 2175 rewritten to use an existential quantifier. One direction of sbalex 2238 without the universal quantifier, avoiding ax-10 2136. (Contributed by SN, 14-Aug-2025.) |
Ref | Expression |
---|---|
ax12ev2 | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnalimn 1842 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) | |
2 | ax12v2 2175 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))) | |
3 | 2 | con1d 145 | . . 3 ⊢ (𝑥 = 𝑦 → (¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) → 𝜑)) |
4 | 1, 3 | biimtrid 242 | . 2 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑)) |
5 | 4 | com12 32 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → 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: sbalex 2238 mopick 2622 sbalexi 42155 |
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