Theorem nfsbc1 3761

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 3760	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑥]𝜑)
4	3	mptru 1549	1 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1543  Ⅎwnf 1785  Ⅎwnfc 2884  [wsbc 3742

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-sbc 3743

This theorem is referenced by:  nfsbc1v  3762  riotass2  7355  riotass  7356  uzwo4  45413