Theorem csbied 3171

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbied.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
csbied.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbied	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbied

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcvd 2373	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐶)
3		csbied.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		csbied.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
5	1, 2, 3, 4	csbiedf 3165	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ⦋csb 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbied2  3172  rspc2vd  3193  fvmptd  5708  seq3f1olemp  10724  fsumgcl  11883  fsum3  11884  fsumshftm  11942  fisum0diag2  11944  fprodseq  12080  fprodeq0  12114  imasival  13325  mulgfvalg  13644  znval  14585  psrval  14615  mplvalcoe  14639  fsumdvdsmul  15650