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| Mirrors > Home > MPE Home > Th. List > exsbim | Structured version Visualization version GIF version | ||
| Description: One direction of the equivalence in exsb 2362 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.) |
| Ref | Expression |
|---|---|
| exsbim | ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | alequexv 2000 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | |
| 2 | 1 | exlimiv 1930 | 1 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 |
| This theorem is referenced by: spsbe 2082 eu6 2574 eu6im 2575 eu6w 42686 |
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