Theorem nfsbd 1906

Description: Deduction version of nfsb 1877. (Contributed by NM, 15-Feb-2013.)

Hypotheses

Ref	Expression
nfsbd.1	⊢ Ⅎ𝑥𝜑
nfsbd.2	⊢ (𝜑 → Ⅎ𝑧𝜓)

Assertion

Ref	Expression
nfsbd	⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Distinct variable group:   𝑦,𝑧

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

Proof of Theorem nfsbd

Step	Hyp	Ref	Expression
1		nfsbd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1464	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfsbd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑧𝜓)
4	3	alimi 1396	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥Ⅎ𝑧𝜓)
5		nfsbt 1905	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
6	2, 4, 5	3syl 17	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1294  Ⅎwnf 1401  [wsb 1699

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700

This theorem is referenced by:  nfeud  1971  nfabd  2254  nfraldya  2423  nfrexdya  2424  cbvrald  12396