Theorem hbsb 2523

Description: If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.) Usage of this theorem is discouraged because it depends on ax-13 2371. Use hbsbw 2169 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
hbsb.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
hbsb	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbsb

Step	Hyp	Ref	Expression
1		hbsb.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nf5i 2142	. . 3 ⊢ Ⅎ𝑧𝜑
3	2	nfsb 2522	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
4	3	nf5ri 2188	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1539  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068

This theorem is referenced by:  hbabg  2721  hblemg  2866