Theorem hbsb 2515

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 2363. Use hbsbw 2161 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 2134	. . 3 ⊢ Ⅎ𝑧𝜑
3	2	nfsb 2514	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
4	3	nf5ri 2180	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1531  [wsb 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060

This theorem is referenced by:  hbabg  2713  hblemg  2858