Theorem r19.41v 2661

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 1550	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2660	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2484

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-rex 2489

This theorem is referenced by:  r19.42v  2662  3reeanv  2676  reuind  2977  iuncom4  3933  dfiun2g  3958  iunxiun  4008  inuni  4198  xpiundi  4731  xpiundir  4732  imaco  5185  coiun  5189  abrexco  5818  imaiun  5819  isoini  5877  rexrnmpo  6051  mapsnen  6888  genpassl  7619  genpassu  7620  4fvwrd4  10244  4sqlem12  12644  metrest  14896  trirec0xor  15848