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Theorem bj-exlimmp 11099
Description: Lemma for bj-vtoclgf 11105. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-exlimmp.nf 𝑥𝜓
bj-exlimmp.min (𝜒𝜑)
Assertion
Ref Expression
bj-exlimmp (∀𝑥(𝜒 → (𝜑𝜓)) → (∃𝑥𝜒𝜓))

Proof of Theorem bj-exlimmp
StepHypRef Expression
1 nfa1 1477 . 2 𝑥𝑥(𝜒 → (𝜑𝜓))
2 bj-exlimmp.nf . 2 𝑥𝜓
3 bj-exlimmp.min . . . . 5 (𝜒𝜑)
4 idd 21 . . . . 5 (𝜒 → (𝜓𝜓))
53, 4embantd 55 . . . 4 (𝜒 → ((𝜑𝜓) → 𝜓))
65a2i 11 . . 3 ((𝜒 → (𝜑𝜓)) → (𝜒𝜓))
76sps 1473 . 2 (∀𝑥(𝜒 → (𝜑𝜓)) → (𝜒𝜓))
81, 2, 7exlimd 1531 1 (∀𝑥(𝜒 → (𝜑𝜓)) → (∃𝑥𝜒𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  wnf 1392  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  bj-vtoclgft  11104
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