Theorem r19.41v 2687

Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)

Assertion

Ref	Expression
r19.41v	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.41v

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2686	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2509

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-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-rex 2514

This theorem is referenced by:  r19.42v  2688  3reeanv  2702  reuind  3008  iuncom4  3971  dfiun2g  3996  iunxiun  4046  inuni  4238  xpiundi  4776  xpiundir  4777  imaco  5233  coiun  5237  abrexco  5882  imaiun  5883  isoini  5941  rexrnmpo  6119  mapsnen  6962  genpassl  7707  genpassu  7708  4fvwrd4  10332  4sqlem12  12920  metrest  15174  trirec0xor  16372