Theorem moimi 2123

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 2122	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
2		moimi.1	. 2 ⊢ (𝜑 → 𝜓)
3	1, 2	mpg 1477	1 ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∃*wmo 2058

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061

This theorem is referenced by:  moan  2127  moor  2129  mooran1  2130  mooran2  2131  2moex  2144  2euex  2145  2exeu  2150  mosubt  2960  sndisj  4058  disjxsn  4060  mosubopt  4761  fununmo  5339  funcnvsn  5342  nfunsn  5638  th3qlem2  6755  shftfn  11301