Theorem euimmo 2522

Description: Uniqueness implies "at most one" 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 2499	. 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓)
2		moim 2519	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
3	1, 2	syl5 34	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1481  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  euim  2523  2eumo  2545  moeq3  3383  reuss2  3907