Theorem r19.41v 2663

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 1552	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2662	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2486

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-rex 2491

This theorem is referenced by:  r19.42v  2664  3reeanv  2678  reuind  2982  iuncom4  3940  dfiun2g  3965  iunxiun  4015  inuni  4207  xpiundi  4741  xpiundir  4742  imaco  5197  coiun  5201  abrexco  5841  imaiun  5842  isoini  5900  rexrnmpo  6074  mapsnen  6917  genpassl  7657  genpassu  7658  4fvwrd4  10282  4sqlem12  12800  metrest  15053  trirec0xor  16125