Theorem mosub 3676

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 3670	. 2 ⊢ ∃*𝑦 𝑦 = 𝐴
2		mosub.1	. . 3 ⊢ ∃*𝑥𝜑
3	2	ax-gen 1825	. 2 ⊢ ∀𝑦∃*𝑥𝜑
4		moexexvw 2656	. 2 ⊢ ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
5	1, 3, 4	mp2an 704	1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)

This proof depends on syntax axioms:   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809  ∃*wmo 2565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-mo 2567  df-cleq 2755

This theorem is used by: (None)