Theorem moa1 2521

Description: If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1741 and exa1 1765. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)

Assertion

Ref	Expression
moa1	⊢ (∃*𝑥(𝜑 → 𝜓) → ∃*𝑥𝜓)

Proof of Theorem moa1

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (𝜓 → (𝜑 → 𝜓))
2	1	moimi 2520	1 ⊢ (∃*𝑥(𝜑 → 𝜓) → ∃*𝑥𝜓)

Syntax hints:   → wi 4  ∃*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: (None)