Theorem moanimv 2158

Description: Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.)

Assertion

Ref	Expression
moanimv	⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem moanimv

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜑
2	1	moanim 2157	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃*wmo 2083

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086

This theorem is referenced by:  mosubt  2997  2reuswapdc  3024  2rmorex  3026  mosubopt  4820  funmo  5372  funcnv  5422  fncnv  5427  isarep2  5448  fnres  5480  fnopabg  5487  fvopab3ig  5756  opabex  5915  fnoprabg  6162  ovidi  6180  ovig  6183  oprabexd  6333  oprabex  6334  th3qcor  6886  dvfgg  15665