Theorem excomim 1593

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 1522	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2	1	2eximi 1532	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦∃𝑥𝜑)
3		hbe1 1424	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
4	3	hbex 1567	. . 3 ⊢ (∃𝑦∃𝑥𝜑 → ∀𝑥∃𝑦∃𝑥𝜑)
5	4	19.9h 1574	. 2 ⊢ (∃𝑥∃𝑦∃𝑥𝜑 ↔ ∃𝑦∃𝑥𝜑)
6	2, 5	sylib 120	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  excom  1594  2euswapdc  2032