Theorem nfsbc1 3794

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 3793	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑥]𝜑)
4	3	mptru 1543	1 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1537  Ⅎwnf 1783  Ⅎwnfc 2964  [wsbc 3775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-sbc 3776

This theorem is referenced by:  nfsbc1v  3795  riotass2  7147  riotass  7148  uzwo4  41321