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  2847  euxfr2dc  2988  euind  2990  sbccomlem  3103  opelopabsbALT  4346  uniuni  4541  elvvv  4781  elco  4887  dmuni  4932  dm0rn0  4939  dmmrnm  4942  dmcosseq  4995  elres  5040  rnco  5234  coass  5246  oprabid  6032  dfoprab2  6050  opabex3d  6264  opabex3  6265  cnvoprab  6378  domen  6898  xpassen  6985  prarloc  7686  fisumcom2  11944  fprodcom2fi  12132