Theorem nfsbc 3741

Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfsbcw 3738 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 1807	. . 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 3740	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1546	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1540  Ⅎwnf 1786  Ⅎwnfc 2887  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-sbc 3717

This theorem is referenced by:  cbvralcsf  3877  ralrnmpt  6972  elovmporab1  7517  opreu2reuALT  30825