Theorem excom 1712

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 1711	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
2		excomim 1711	. 2 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)
3	1, 2	impbii 126	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

Syntax hints:   ↔ wb 105  ∃wex 1540

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  excom13  1737  exrot3  1738  ee4anv  1987  sbexyz  2056  2exsb  2062  2euex  2167  2exeu  2172  2eu4  2173  rexcomf  2695  gencbvex  2850  euxfr2dc  2991  euind  2993  sbccomlem  3106  opelopabsbALT  4353  uniuni  4548  elvvv  4789  elco  4896  dmuni  4941  dm0rn0  4948  dmmrnm  4951  dmcosseq  5004  elres  5049  rnco  5243  coass  5255  oprabid  6049  dfoprab2  6067  opabex3d  6282  opabex3  6283  cnvoprab  6398  domen  6921  xpassen  7013  prarloc  7722  fisumcom2  11998  fprodcom2fi  12186