Theorem r19.41v 2650

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 1539	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2649	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-rex 2478

This theorem is referenced by:  r19.42v  2651  3reeanv  2665  reuind  2965  iuncom4  3919  dfiun2g  3944  iunxiun  3994  inuni  4184  xpiundi  4717  xpiundir  4718  imaco  5171  coiun  5175  abrexco  5802  imaiun  5803  isoini  5861  rexrnmpo  6034  mapsnen  6865  genpassl  7584  genpassu  7585  4fvwrd4  10206  4sqlem12  12540  metrest  14674  trirec0xor  15535