Theorem nfsbd 2530

Description: Deduction version of nfsb 2531. (Contributed by NM, 15-Feb-2013.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use nfsbv 2339 instead. (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 2225	. . 3 ⊢ (𝜑 → ∀𝑥Ⅎ𝑧𝜓)
4		nfsb4t 2507	. . 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 2275	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
7	5, 6	pm2.61d2 182	1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545  Ⅎwnf 1790  [wsb 2073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074

This theorem is referenced by:  nfsb  2531  nfabd  2924  wl-sb8eut  37956