Theorem nfsbd 1970

Description: Deduction version of nfsb 1939. (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 1512	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfsbd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑧𝜓)
4	3	alimi 1448	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥Ⅎ𝑧𝜓)
5		nfsbt 1969	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
6	2, 4, 5	3syl 17	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1346  Ⅎwnf 1453  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  nfeud  2035  nfabd  2332  nfraldya  2505  nfrexdya  2506  cbvrald  13823