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Theorem nfsb 2428
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfsb.1 𝑧𝜑
Assertion
Ref Expression
nfsb 𝑧[𝑦 / 𝑥]𝜑
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsb
StepHypRef Expression
1 axc16nf 2122 . 2 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
2 nfsb.1 . . 3 𝑧𝜑
32nfsb4 2378 . 2 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
41, 3pm2.61i 175 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wal 1473  wnf 1699  [wsb 1867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868
This theorem is referenced by:  hbsb  2429  sb10f  2444  2sb8e  2455  sb8eu  2491  2mo  2539  cbvralf  3141  cbvreu  3145  cbvralsv  3158  cbvrexsv  3159  cbvrab  3171  cbvreucsf  3533  cbvrabcsf  3534  cbvopab1  4655  cbvmptf  4676  cbvmpt  4677  ralxpf  5190  cbviota  5773  sb8iota  5775  cbvriota  6521  dfoprab4f  7117  mo5f  28708  ax11-pm2  32011  2sb5nd  37797
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