Theorem excomim 1689

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 1616	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2	1	2eximi 1627	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦∃𝑥𝜑)
3		hbe1 1521	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
4	3	hbex 1662	. . 3 ⊢ (∃𝑦∃𝑥𝜑 → ∀𝑥∃𝑦∃𝑥𝜑)
5	4	19.9h 1669	. 2 ⊢ (∃𝑥∃𝑦∃𝑥𝜑 ↔ ∃𝑦∃𝑥𝜑)
6	2, 5	sylib 122	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1518

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 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-4 1536  ax-ial 1560

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  excom  1690  2euswapdc  2149