Theorem nfsbd 1988

Description: Deduction version of nfsb 1957. (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 1529	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfsbd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑧𝜓)
4	3	alimi 1465	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥Ⅎ𝑧𝜓)
5		nfsbt 1987	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
6	2, 4, 5	3syl 17	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1361  Ⅎwnf 1470  [wsb 1772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773

This theorem is referenced by:  nfeud  2053  nfabd  2351  nfraldya  2524  nfrexdya  2525  cbvrald  14923