Theorem r19.41v 2626

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 1521	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2625	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-rex 2454

This theorem is referenced by:  r19.42v  2627  3reeanv  2640  reuind  2935  iuncom4  3880  dfiun2g  3905  iunxiun  3954  inuni  4141  xpiundi  4669  xpiundir  4670  imaco  5116  coiun  5120  abrexco  5738  imaiun  5739  isoini  5797  rexrnmpo  5968  mapsnen  6789  genpassl  7486  genpassu  7487  4fvwrd4  10096  metrest  13300  trirec0xor  14077