Theorem nfsb 2555

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 2363 for a version with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2404. (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 2404. Use nfsbv 2363 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 1834	. . 3 ⊢ Ⅎ𝑥⊤
2		nfsb.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
4	1, 3	nfsbd 2554	. 2 ⊢ (⊤ → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
5	4	mptru 1577	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1571  Ⅎwnf 1813  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by:  hbsb  2556  sb10f  2559  2sb8e  2562  sb8eu  2628  cbvralf  3349  cbvralsv  3355  cbvrexsv  3356  cbvreu  3408  cbvrab  3454  cbvreucsf  3898  cbvrabcsf  3899  cbvopab1g  5187  cbvmptfg  5213  cbviota  6503  sb8iota  6505  cbvriota  7382  2sb5nd  45252