Theorem nfsbd 2541

Description: Deduction version of nfsb 2542. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 15-Feb-2013.) (New usage is discouraged.)

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	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜓)
3	1, 2	alrimi 2211	. . 3 ⊢ (𝜑 → ∀𝑥Ⅎ𝑧𝜓)
4		nfsb4t 2517	. . 3 ⊢ (∀𝑥Ⅎ𝑧𝜓 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓))
6		axc16nf 2261	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
7	5, 6	pm2.61d2 184	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  Ⅎwnf 1785  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070

This theorem is referenced by:  nfsb  2542  nfabd  2977  wl-sb8eut  34978