Theorem nfcsb 3949

Description: Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfcsbw 3948 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfcsb.1	⊢ Ⅎ𝑥𝐴
nfcsb.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfcsb	⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Proof of Theorem nfcsb

Step	Hyp	Ref	Expression
1		nftru 1802	. . 3 ⊢ Ⅎ𝑦⊤
2		nfcsb.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfcsb.2	. . . 4 ⊢ Ⅎ𝑥𝐵
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐵)
6	1, 3, 5	nfcsbd 3947	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
7	6	mptru 1544	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1538  Ⅎwnfc 2893  ⦋csb 3921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-sbc 3805  df-csb 3922

This theorem is referenced by:  cbvralcsf  3966  cbvreucsf  3968  cbvrabcsf  3969  elfvmptrab1  7057  elovmporab1  7698