Theorem 19.41vv 1918

Description: Version of 19.41 2141 with two quantifiers and a dv 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 1917	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ 𝜓))
2	1	exbii 1814	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜓))
3		19.41v 1917	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))
4	2, 3	bitri 264	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 196   ∧ wa 383  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745

This theorem is referenced by:  19.41vvv  1919  rabxp  5188  copsex2gb  5263  mpt2mptx  6793  xpassen  8095  dfac5lem1  8984  fusgr2wsp2nb  27314  bnj996  31151  dfdm5  31800  dfrn5  31801  elima4  31803  brtxp2  32113  brpprod3a  32118  brimg  32169  brsuccf  32173  brxrn2  34277  diblsmopel  36777  mpt2mptx2  42438