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Theorem mosub 3542
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 3534 . 2 ∃*𝑦 𝑦 = 𝐴
2 mosub.1 . . 3 ∃*𝑥𝜑
32ax-gen 1890 . 2 𝑦∃*𝑥𝜑
4 moexexv 2663 . 2 ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
51, 3, 4mp2an 683 1 ∃*𝑥𝑦(𝑦 = 𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wal 1650   = wceq 1652  wex 1874  ∃*wmo 2562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-cleq 2757
This theorem is referenced by: (None)
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