Theorem nfexd 1854

Description: If x is not free in φ, it is not free in ∃yφ. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
nfald.1	⊢ Ⅎyφ
nfald.2	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfexd	⊢ (φ → Ⅎx∃yψ)

Proof of Theorem nfexd

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 ⊢ (∃yψ ↔ ¬ ∀y ¬ ψ)
2		nfald.1	. . . 4 ⊢ Ⅎyφ
3		nfald.2	. . . . 5 ⊢ (φ → Ⅎxψ)
4	3	nfnd 1791	. . . 4 ⊢ (φ → Ⅎx ¬ ψ)
5	2, 4	nfald 1852	. . 3 ⊢ (φ → Ⅎx∀y ¬ ψ)
6	5	nfnd 1791	. 2 ⊢ (φ → Ⅎx ¬ ∀y ¬ ψ)
7	1, 6	nfxfrd 1571	1 ⊢ (φ → Ⅎx∃yψ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

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  df-nf 1545

This theorem is referenced by:  nfeld  2505