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Theorem bj-exlimmp 13145
Description: Lemma for bj-vtoclgf 13152. (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 1522 . 2 𝑥𝑥(𝜒 → (𝜑𝜓))
2 bj-exlimmp.nf . 2 𝑥𝜓
3 bj-exlimmp.min . . . . 5 (𝜒𝜑)
4 idd 21 . . . . 5 (𝜒 → (𝜓𝜓))
53, 4embantd 56 . . . 4 (𝜒 → ((𝜑𝜓) → 𝜓))
65a2i 11 . . 3 ((𝜒 → (𝜑𝜓)) → (𝜒𝜓))
76sps 1518 . 2 (∀𝑥(𝜒 → (𝜑𝜓)) → (𝜒𝜓))
81, 2, 7exlimd 1577 1 (∀𝑥(𝜒 → (𝜑𝜓)) → (∃𝑥𝜒𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1330  wnf 1437  wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  bj-vtoclgft  13151
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