Theorem mosub 2958

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 2957	. 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
2		mosub.1	. 2 ⊢ ∃*𝑥𝜑
3	1, 2	mpg 1475	1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516  ∃*wmo 2056

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by: (None)