Theorem nfcsb1 3916

Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfcsb1.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcsb1	⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵

Proof of Theorem nfcsb1

Step	Hyp	Ref	Expression
1		nfcsb1.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
3	2	nfcsb1d 3915	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
4	3	mptru 1541	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ⊤wtru 1535  Ⅎwnfc 2876  ⦋csb 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-sbc 3777  df-csb 3893

This theorem is referenced by:  nfcsb1v  3917  fsumsplit1  15742  iundisj  25563  disjabrex  32500  disjabrexf  32501  iundisjf  32507  iundisjfi  32699  rdgssun  37096  evl1gprodd  41827  disjinfi  44833  fsumsermpt  45234  climsubmpt  45315  climeldmeqmpt  45323  climfveqmpt  45326  climfveqmpt3  45337  climeldmeqmpt3  45344  climinf2mpt  45369  climinfmpt  45370  dvmptmulf  45592  dvnmptdivc  45593  sge0f1o  46037  sge0lempt  46065  sge0isummpt2  46087  meadjiun  46121  hoimbl2  46320  vonhoire  46327