Theorem 19.12 1537

Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.12	⊢ (∃x∀yφ → ∀y∃xφ)

Proof of Theorem 19.12

Step	Hyp	Ref	Expression
1		hba1 1415	. . 3 ⊢ (∀yφ → ∀y∀yφ)
2	1	hbex 1509	. 2 ⊢ (∃x∀yφ → ∀y∃x∀yφ)
3		ax-4 1381	. . 3 ⊢ (∀yφ → φ)
4	3	eximi 1473	. 2 ⊢ (∃x∀yφ → ∃xφ)
5	2, 4	alrimih 1338	1 ⊢ (∃x∀yφ → ∀y∃xφ)

Syntax hints:   → wi 4  ∀wal 1226  ∃wex 1362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  hbexd  1566  nfexd  1626  cbvexdh  1783