Theorem nfsbc 3764

Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfsbcw 3761 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 1806	. . 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 3763	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1549	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1543  Ⅎwnf 1785  Ⅎwnfc 2882  [wsbc 3739

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 2183  ax-13 2375  ax-ext 2707

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 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-sbc 3740

This theorem is referenced by:  cbvralcsf  3890  ralrnmpt  7041  elovmporab1  7606  opreu2reuALT  32531