Theorem nfsbc1 3763

Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfsbc1.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfsbc1	⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑

Proof of Theorem nfsbc1

Step	Hyp	Ref	Expression
1		nfsbc1.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
3	2	nfsbc1d 3762	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑥]𝜑)
4	3	mptru 1577	1 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑

This proof depends on syntax axioms:  ⊤wtru 1571  Ⅎwnf 1813  Ⅎwnfc 2910  [wsbc 3744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-sbc 3745

This theorem is used by:  nfsbc1v  3764  riotass2  7397  riotass  7398  uzwo4  45801