Theorem nfsbd 2577

Description: Deduction version of nfsb 2575. (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	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜓)
3	1, 2	alrimi 2256	. . 3 ⊢ (𝜑 → ∀𝑥Ⅎ𝑧𝜓)
4		nfsb4t 2520	. . 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 2293	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
7	5, 6	pm2.61d2 174	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1654  Ⅎwnf 1882  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068

This theorem is referenced by:  nfabd2  2989  wl-sb8eut  33902