Theorem 19.12 1689

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 1564	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑦𝜑)
2	1	hbex 1660	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥∀𝑦𝜑)
3		ax-4 1534	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1624	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimih 1493	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1371  ∃wex 1516

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbexd  1718  nfexd  1785  cbvexdh  1951