Theorem excom 1710

Description: Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
excom	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Proof of Theorem excom

Step	Hyp	Ref	Expression
1		excomim 1709	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
2		excomim 1709	. 2 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)
3	1, 2	impbii 126	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Syntax hints:   ↔ wb 105  ∃wex 1538

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  excom13  1735  exrot3  1736  ee4anv  1985  sbexyz  2054  2exsb  2060  2euex  2165  2exeu  2170  2eu4  2171  rexcomf  2693  gencbvex  2848  euxfr2dc  2989  euind  2991  sbccomlem  3104  opelopabsbALT  4351  uniuni  4546  elvvv  4787  elco  4894  dmuni  4939  dm0rn0  4946  dmmrnm  4949  dmcosseq  5002  elres  5047  rnco  5241  coass  5253  oprabid  6045  dfoprab2  6063  opabex3d  6278  opabex3  6279  cnvoprab  6394  domen  6917  xpassen  7009  prarloc  7713  fisumcom2  11989  fprodcom2fi  12177