Theorem nfcsb1 3873

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 3872	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
4	3	mptru 1577	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵

This proof depends on syntax axioms:  ⊤wtru 1571  Ⅎwnfc 2909  ⦋csb 3850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-sbc 3743  df-csb 3851

This theorem is used by:  nfcsb1v  3874  fsumsplit1  15835  iundisj  25782  disjabrex  33063  disjabrexf  33064  iundisjf  33070  iundisjfi  33275  rdgssun  38140  evl1gprodd  42991  disjinfi  46032  fsumsermpt  46417  climsubmpt  46496  climeldmeqmpt  46504  climfveqmpt  46507  climfveqmpt3  46518  climeldmeqmpt3  46525  climinf2mpt  46550  climinfmpt  46551  dvmptmulf  46773  dvnmptdivc  46774  sge0lempt  47246  sge0isummpt2  47268  meadjiun  47302  hoimbl2  47501  vonhoire  47508