Theorem nfsbc 3776

Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2410. Use the weaker nfsbcw 3773 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfsbc.1	⊢ Ⅎ𝑥𝐴
nfsbc.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfsbc	⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Proof of Theorem nfsbc

Step	Hyp	Ref	Expression
1		nftru 1831	. . 3 ⊢ Ⅎ𝑦⊤
2		nfsbc.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfsbc.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfsbcd 3775	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1574	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1568  Ⅎwnf 1810  Ⅎwnfc 2916  [wsbc 3751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-sbc 3752

This theorem is referenced by:  cbvralcsf  3901  ralrnmpt  7092  elovmporab1  7659  opreu2reuALT  32764