Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-alequex | Structured version Visualization version GIF version |
Description: A fol lemma. Can be used to reduce the proof of spimt 2241 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 2238 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1751 | . 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 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: bj-spimt2 31896 bj-equsal1t 31997 |
Copyright terms: Public domain | W3C validator |