Theorem moimi 2144

Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Hypothesis

Ref	Expression
moimi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
moimi	⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Proof of Theorem moimi

Step	Hyp	Ref	Expression
1		moim 2143	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
2		moimi.1	. 2 ⊢ (𝜑 → 𝜓)
3	1, 2	mpg 1499	1 ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∃*wmo 2079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082

This theorem is referenced by:  moan  2148  moor  2150  mooran1  2151  mooran2  2152  2moex  2165  2euex  2166  2exeu  2171  mosubt  2982  sndisj  4085  disjxsn  4087  mosubopt  4793  fununmo  5374  funcnvsn  5377  nfunsn  5679  th3qlem2  6812  shftfn  11407