Theorem euimmo 2644

Description: Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
euimmo	⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Proof of Theorem euimmo

Step	Hyp	Ref	Expression
1		eumo 2606	. 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓)
2		moim 2572	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
3	1, 2	syl5 35	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃*wmo 2565  ∃!weu 2596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2567  df-eu 2597

This theorem is used by:  euim  2645  2eumo  2670  moeq3  3673  reuss2  4274