MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mosub Structured version   Visualization version   GIF version

Theorem mosub 3708
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 3702 . 2 ∃*𝑦 𝑦 = 𝐴
2 mosub.1 . . 3 ∃*𝑥𝜑
32ax-gen 1789 . 2 𝑦∃*𝑥𝜑
4 moexexvw 2712 . 2 ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
51, 3, 4mp2an 688 1 ∃*𝑥𝑦(𝑦 = 𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wal 1528   = wceq 1530  wex 1773  ∃*wmo 2618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-cleq 2819
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator