Theorem hbex 1841

Description: If x is not free in φ, it is not free in ∃yφ. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbex.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbex	⊢ (∃yφ → ∀x∃yφ)

Proof of Theorem hbex

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 ⊢ (∃yφ ↔ ¬ ∀y ¬ φ)
2		hbex.1	. . . . 5 ⊢ (φ → ∀xφ)
3	2	hbn 1776	. . . 4 ⊢ (¬ φ → ∀x ¬ φ)
4	3	hbal 1736	. . 3 ⊢ (∀y ¬ φ → ∀x∀y ¬ φ)
5	4	hbn 1776	. 2 ⊢ (¬ ∀y ¬ φ → ∀x ¬ ∀y ¬ φ)
6	1, 5	hbxfrbi 1568	1 ⊢ (∃yφ → ∀x∃yφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  nfex  1843  19.12OLD  1848  hboprab2  5560  hboprab3  5561  hboprab  5562