Theorem mosub 2930

Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)

Hypothesis

Ref	Expression
mosub.1	⊢ ∃*𝑥𝜑

Assertion

Ref	Expression
mosub	⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mosub

Step	Hyp	Ref	Expression
1		mosubt 2929	. 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
2		mosub.1	. 2 ⊢ ∃*𝑥𝜑
3	1, 2	mpg 1462	1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503  ∃*wmo 2039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754

This theorem is referenced by: (None)