Theorem csbeq2dv 4025

Description: Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Hypothesis

Ref	Expression
csbeq2dv.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbeq2dv	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbeq2dv

Step	Hyp	Ref	Expression
1		nfv 1883	. 2 ⊢ Ⅎ𝑥𝜑
2		csbeq2dv.1	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
3	1, 2	csbeq2d 4024	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1523  ⦋csb 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-sbc 3469  df-csb 3567

This theorem is referenced by:  csbeq2i  4026  mpt2mptsx  7278  dmmpt2ssx  7280  fmpt2x  7281  el2mpt2csbcl  7295  offval22  7298  ovmptss  7303  fmpt2co  7305  mpt2sn  7313  mpt2curryd  7440  fvmpt2curryd  7442  cantnffval  8598  fsumcom2  14549  fsumcom2OLD  14550  fprodcom2  14758  fprodcom2OLD  14759  bpolylem  14823  bpolyval  14824  ruclem1  15004  natfval  16653  fucval  16665  evlfval  16904  mpfrcl  19566  pmatcollpw3lem  20636  fsumcn  22720  fsum2cn  22721  dvmptfsum  23783  ttgval  25800  msrfval  31560  poimirlem5  33544  poimirlem6  33545  poimirlem7  33546  poimirlem8  33547  poimirlem10  33549  poimirlem11  33550  poimirlem12  33551  poimirlem15  33554  poimirlem16  33555  poimirlem17  33556  poimirlem18  33557  poimirlem19  33558  poimirlem20  33559  poimirlem21  33560  poimirlem22  33561  poimirlem24  33563  poimirlem26  33565  poimirlem27  33566  cdleme31sde  35990  cdlemeg47rv2  36115  rnghmval  42216  dmmpt2ssx2  42440