Theorem nfsb 2525

Description: If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. See nfsbv 2333 for a version with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.) Usage of this theorem is discouraged because it depends on ax-13 2374. Use nfsbv 2333 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
nfsb.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsb	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable group:   𝑦,𝑧

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

Proof of Theorem nfsb

Step	Hyp	Ref	Expression
1		nftru 1805	. . 3 ⊢ Ⅎ𝑥⊤
2		nfsb.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
4	1, 3	nfsbd 2524	. 2 ⊢ (⊤ → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
5	4	mptru 1548	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1542  Ⅎwnf 1784  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  hbsb  2526  sb10f  2529  2sb8e  2532  sb8eu  2597  cbvralf  3327  cbvralsv  3333  cbvrexsv  3334  cbvreu  3388  cbvrab  3436  cbvreucsf  3890  cbvrabcsf  3891  cbvopab1g  5170  cbvmptfg  5196  cbviota  6454  sb8iota  6456  cbvriota  7325  2sb5nd  44717