Theorem r19.41v 2699

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 2698	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2521

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 2526

This theorem is referenced by:  r19.42v  2700  3reeanv  2714  reuind  3022  iuncom4  3998  dfiun2g  4023  iunxiun  4073  inuni  4267  xpiundi  4808  xpiundir  4809  imaco  5268  coiun  5272  abrexco  5932  imaiun  5933  isoini  5991  rexrnmpo  6169  mapsnend  7052  mapsnen  7053  genpassl  7839  genpassu  7840  4fvwrd4  10474  4sqlem12  13100  metrest  15371  trirec0xor  16829