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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 2382 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 | ax6ev 2077 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1932 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
4 | 3 | exlimiv 2029 | 1 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1654 ∃wex 1878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 |
This theorem depends on definitions: df-bi 199 df-ex 1879 |
This theorem is referenced by: eu6im 2646 |
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