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Mirrors > Home > MPE Home > Th. List > alequexv | Structured version Visualization version GIF version |
Description: Version of equs4v 2006 with its consequence simplified by exsimpr 1875. (Contributed by BJ, 9-Nov-2021.) |
Ref | Expression |
---|---|
alequexv | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6ev 1976 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1839 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∃wex 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-6 1974 |
This theorem depends on definitions: df-bi 206 df-ex 1786 |
This theorem is referenced by: exsbim 2008 spsbe 2088 19.8a 2177 bj-spimvwt 34829 |
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