Theorem hbex 2355

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 2354, hbex 2355. (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 2182	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfex 2354	. 2 ⊢ Ⅎ𝑥∃𝑦𝜑
4	3	nf5ri 2231	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by: (None)