Theorem euimmo 2620

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 2582	. 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓)
2		moim 2548	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
3	1, 2	syl5 34	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1545  ∃*wmo 2541  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573

This theorem is referenced by:  euim  2621  2eumo  2646  moeq3  3660  reuss2  4261