Theorem 19.41vv 1980

Description: Version of 19.41 2271 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vv	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.41vv

Step	Hyp	Ref	Expression
1		19.41v 1979	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ 𝜓))
2	1	exbii 1878	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜓))
3		19.41v 1979	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))
4	2, 3	bitri 278	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  19.41vvv  1981  cgsex4g  3501  rabxp  5711  copsex2gb  5795  mpomptx  7525  xpassen  9060  dfac5lem1  10108  fusgr2wsp2nb  30663  bnj996  35322  dfdm5  36243  dfrn5  36244  elima4  36246  brtxp2  36349  brpprod3a  36354  brimg  36405  lemsuccf  36409  brxrn2  39011  diblsmopel  41923  en2pr  44253  mpomptx2  49092