Theorem nfsbd 2006

Description: Deduction version of nfsb 1975. (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 1543	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfsbd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑧𝜓)
4	3	alimi 1479	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥Ⅎ𝑧𝜓)
5		nfsbt 2005	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
6	2, 4, 5	3syl 17	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1484  [wsb 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  nfeud  2071  nfabd  2369  nfraldya  2542  nfrexdya  2543  cbvrald  15798