Theorem 19.12 1643

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 1520	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑦𝜑)
2	1	hbex 1615	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥∀𝑦𝜑)
3		ax-4 1487	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	eximi 1579	. 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑)
5	2, 4	alrimih 1445	1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  hbexd  1672  nfexd  1734  cbvexdh  1898