Theorem hbex 2323

Description: If 𝑥 is not free in 𝜑, then it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2322, hbex 2323. (Revised by Wolf Lammen, 16-Oct-2021.)

Hypothesis

Ref	Expression
hbex.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbex	⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Proof of Theorem hbex

Step	Hyp	Ref	Expression
1		hbex.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2144	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfex 2322	. 2 ⊢ Ⅎ𝑥∃𝑦𝜑
4	3	nf5ri 2191	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)