Theorem 19.42vv 1957

Description: Version of 19.42 2237 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)

Assertion

Ref	Expression
19.42vv	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem 19.42vv

Step	Hyp	Ref	Expression
1		exdistr 1954	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
2		19.42v 1953	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))
3	1, 2	bitri 275	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  exdistr2  1958  3exdistr  1960  cgsex4g  3491  ceqsex3v  3500  ceqsex4v  3501  ceqsex8v  3503  elvvv  5707  xpdifid  6129  dfoprab2  7427  resoprab  7487  elrnmpores  7507  ov3  7532  ov6g  7533  oprabex3  7935  xpassen  9012  entrfil  9126  domtrfil  9133  sbthfilem  9139  axaddf  11074  axmulf  11075  catcone0  17624  qqhval2  33945  bnj996  34919  fineqvac  35060  inxpxrn  38354  dmqsblocks  38818  dvhopellsm  41084