Theorem r19.41v 2689

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 1576	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.41 2688	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-rex 2516

This theorem is referenced by:  r19.42v  2690  3reeanv  2704  reuind  3011  iuncom4  3977  dfiun2g  4002  iunxiun  4052  inuni  4245  xpiundi  4784  xpiundir  4785  imaco  5242  coiun  5246  abrexco  5899  imaiun  5900  isoini  5958  rexrnmpo  6136  mapsnen  6985  genpassl  7743  genpassu  7744  4fvwrd4  10374  4sqlem12  12974  metrest  15229  trirec0xor  16649