Theorem 19.41vv 1952

Description: Version of 19.41 2243 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 1951	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ 𝜓))
2	1	exbii 1850	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜓))
3		19.41v 1951	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))
4	2, 3	bitri 275	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  19.41vvv  1953  cgsex4g  3489  rabxp  5680  copsex2gb  5763  mpomptx  7481  xpassen  9011  dfac5lem1  10045  fusgr2wsp2nb  30421  bnj996  35131  dfdm5  35986  dfrn5  35987  elima4  35989  brtxp2  36092  brpprod3a  36097  brimg  36148  lemsuccf  36152  brxrn2  38629  diblsmopel  41541  en2pr  43897  mpomptx2  48689