Theorem moanimv 2117

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 1539	. 2 ⊢ Ⅎ𝑥𝜑
2	1	moanim 2116	1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃*wmo 2043

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046

This theorem is referenced by:  mosubt  2937  2reuswapdc  2964  2rmorex  2966  mosubopt  4724  funmo  5269  funcnv  5315  fncnv  5320  isarep2  5341  fnres  5370  fnopabg  5377  fvopab3ig  5631  opabex  5782  fnoprabg  6019  ovidi  6037  ovig  6040  oprabexd  6179  oprabex  6180  th3qcor  6693  dvfgg  14842