Theorem mosub 3667

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 3661	. 2 ⊢ ∃*𝑦 𝑦 = 𝐴
2		mosub.1	. . 3 ⊢ ∃*𝑥𝜑
3	2	ax-gen 1796	. 2 ⊢ ∀𝑦∃*𝑥𝜑
4		moexexvw 2623	. 2 ⊢ ((∃*𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
5	1, 3, 4	mp2an 692	1 ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)

Syntax hints:   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780  ∃*wmo 2533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-mo 2535  df-cleq 2723

This theorem is referenced by: (None)