Theorem 19.12 1595

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 1473	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑦𝜑)
2	1	hbex 1567	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥∀𝑦𝜑)
3		ax-4 1440	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1531	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimih 1398	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1282  ∃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:  hbexd  1624  nfexd  1684  cbvexdh  1842