Theorem r19.41v 2701

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 1577	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2700	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2523

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-rex 2528

This theorem is referenced by:  r19.42v  2702  3reeanv  2716  reuind  3025  iuncom4  4003  dfiun2g  4028  iunxiun  4078  inuni  4272  xpiundi  4813  xpiundir  4814  imaco  5273  coiun  5277  abrexco  5938  imaiun  5939  isoini  5997  rexrnmpo  6177  mapsnend  7065  mapsnen  7066  genpassl  7855  genpassu  7856  4fvwrd4  10496  4sqlem12  13125  metrest  15497  trirec0xor  16955