Theorem mosub 3015

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		moeq 3013	. 2 ⊢ ∃*y y = A
2		mosub.1	. . 3 ⊢ ∃*xφ
3	2	ax-gen 1546	. 2 ⊢ ∀y∃*xφ
4		moexexv 2274	. 2 ⊢ ((∃*y y = A ∧ ∀y∃*xφ) → ∃*x∃y(y = A ∧ φ))
5	1, 3, 4	mp2an 653	1 ⊢ ∃*x∃y(y = A ∧ φ)

Syntax hints:   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)