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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-alequex | Structured version Visualization version GIF version |
Description: A fol lemma. See bj-alequexv 32655 for a version with a DV condition requiring fewer axioms. Can be used to reduce the proof of spimt 2253 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-alequex | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2250 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1761 | . 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 1481 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: bj-spimt2 32709 bj-equsal1t 32809 |
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