Theorem mosub 1919

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:   x,y,A

Proof of Theorem mosub

Step	Hyp	Ref	Expression
1		moeq 1917	. 2 ⊢ ∃*y y = A
2		mosub.1	. . 3 ⊢ ∃*xφ
3	2	ax-gen 962	. 2 ⊢ ∀y∃*xφ
4		moexexv 1438	. 2 ⊢ ((∃*y y = A ⋀ ∀y∃*xφ) → ∃*x∃y(y = A ⋀ φ))
5	1, 3, 4	mp2an 696	1 ⊢ ∃*x∃y(y = A ⋀ φ)

Syntax hints:   ⋀ wa 223  ∀wal 953   = wceq 955  ∃wex 979  ∃*wmo 1380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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