Theorem r19.41v 2690

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 2689	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2512

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 2517

This theorem is referenced by:  r19.42v  2691  3reeanv  2705  reuind  3012  iuncom4  3982  dfiun2g  4007  iunxiun  4057  inuni  4250  xpiundi  4790  xpiundir  4791  imaco  5249  coiun  5253  abrexco  5910  imaiun  5911  isoini  5969  rexrnmpo  6147  mapsnen  7029  genpassl  7787  genpassu  7788  4fvwrd4  10420  4sqlem12  13038  metrest  15300  trirec0xor  16760