Theorem nfimOLD 1814

Description: If x is not free in φ and ψ, it is not free in (φ → ψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfim.1	⊢ Ⅎxφ
nfim.2	⊢ Ⅎxψ

Assertion

Ref	Expression
nfimOLD	⊢ Ⅎx(φ → ψ)

Proof of Theorem nfimOLD

Step	Hyp	Ref	Expression
1		nfim.1	. . . 4 ⊢ Ⅎxφ
2	1	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎxφ)
3		nfim.2	. . . 4 ⊢ Ⅎxψ
4	3	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎxψ)
5	2, 4	nfimd 1808	. 2 ⊢ ( ⊤ → Ⅎx(φ → ψ))
6	5	trud 1323	1 ⊢ Ⅎx(φ → ψ)

This proof depends on syntax axioms:   → wi 4   ⊤ wtru 1316  Ⅎwnf 1544

This proof depends on 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-11 1746

This proof depends on definitions:  df-bi 177  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by: (None)