Theorem r19.41v 2633

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 1528	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2632	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-rex 2461

This theorem is referenced by:  r19.42v  2634  3reeanv  2648  reuind  2943  iuncom4  3894  dfiun2g  3919  iunxiun  3969  inuni  4156  xpiundi  4685  xpiundir  4686  imaco  5135  coiun  5139  abrexco  5760  imaiun  5761  isoini  5819  rexrnmpo  5990  mapsnen  6811  genpassl  7523  genpassu  7524  4fvwrd4  10140  metrest  14009  trirec0xor  14796