Theorem excomim 1656

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

Assertion

Ref	Expression
excomim	⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Proof of Theorem excomim

Step	Hyp	Ref	Expression
1		19.8a 1583	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2	1	2eximi 1594	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦∃𝑥𝜑)
3		hbe1 1488	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
4	3	hbex 1629	. . 3 ⊢ (∃𝑦∃𝑥𝜑 → ∀𝑥∃𝑦∃𝑥𝜑)
5	4	19.9h 1636	. 2 ⊢ (∃𝑥∃𝑦∃𝑥𝜑 ↔ ∃𝑦∃𝑥𝜑)
6	2, 5	sylib 121	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  excom  1657  2euswapdc  2110