Theorem hbsb 2532

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 2380. Use hbsbw 2172 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 2146	. . 3 ⊢ Ⅎ𝑧𝜑
3	2	nfsb 2531	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
4	3	nf5ri 2196	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by:  hbabg  2729  hblemg  2877