Theorem nfsbc1 3810

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 3809	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑥]𝜑)
4	3	mptru 1544	1 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1538  Ⅎwnf 1780  Ⅎwnfc 2888  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792

This theorem is referenced by:  nfsbc1v  3811  riotass2  7418  riotass  7419  uzwo4  44993