Theorem mosub 1433

Description: "At most one" remains true after substitution.

Hypothesis

Ref	Expression
mosub.1	⊢ ∃*xφ

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem mosub

Step	Hyp	Ref	Expression
1		moeq 1431	. 2 ⊢ ∃*y y = A
2		mosub.1	. . 3 ⊢ ∃*xφ
3	2	ax-gen 677	. 2 ⊢ ∀y∃*xφ
4		moexexv 1059	. 2 ⊢ ((∃*y y = A ∧ ∀y∃*xφ) → ∃*x∃y(y = A ∧ φ))
5	1, 3, 4	mp2an 520	1 ⊢ ∃*x∃y(y = A ∧ φ)

Syntax hints:   ∧ wa 196  ∀wal 672  ∃wex 678  ∃*wmo 1008   = wceq 1091

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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