Theorem mosub 2719

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 2718	. 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
2		mosub.1	. 2 ⊢ ∃*𝑥𝜑
3	1, 2	mpg 1340	1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)

Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381  ∃*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)