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Theorem mosub 3378
Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
Hypothesis
Ref Expression
mosub.1 ∃*𝑥𝜑
Assertion
Ref Expression
mosub ∃*𝑥𝑦(𝑦 = 𝐴𝜑)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mosub
StepHypRef Expression
1 moeq 3376 . 2 ∃*𝑦 𝑦 = 𝐴
2 mosub.1 . . 3 ∃*𝑥𝜑
32ax-gen 1720 . 2 𝑦∃*𝑥𝜑
4 moexexv 2540 . 2 ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
51, 3, 4mp2an 707 1 ∃*𝑥𝑦(𝑦 = 𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wal 1479   = wceq 1481  wex 1702  ∃*wmo 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197
This theorem is referenced by: (None)
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