Theorem csbeq2dv 3858

Description: Formula-building deduction 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

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq2dv.1	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	eleq2d 2850	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
3	2	sbcbidv 3797	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
4	3	abbidv 2829	. 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
5		df-csb 3852	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		df-csb 3852	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2823	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741  [wsbc 3742  ⦋csb 3851

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743  df-csb 3852

This theorem is used by:  csbeq2i  3859  csbeq12dv  3860  mpomptsx  8100  dmmpossx  8102  fmpox  8103  el2mpocsbcl  8121  offval22  8124  ovmptssOLD  8130  fmpoco  8132  mposn  8141  mpocurryd  8314  fvmpocurryd  8316  cantnffval  9691  sumeq2sdv  15899  fsumcom2  15969  prodeq2sdv  16120  fprodcom2  16180  bpolylem  16243  bpolyval  16244  ruclem1  16428  natfval  18153  fucval  18165  evlfval  18420  rnghmval  20700  rhmval0  20735  mpfrcl  22424  selvffval  22457  selvfval  22458  selvval  22459  pmatcollpw3lem  23131  fsumcn  25221  fsum2cn  25222  itgeq1f  26122  itgeq1  26123  dvmptfsum  26325  mulsval  28525  precsexlemcbv  28622  msrfval  36469  nmulprop  37117  poimirlem5  38735  poimirlem6  38736  poimirlem7  38737  poimirlem8  38738  poimirlem10  38740  poimirlem11  38741  poimirlem12  38742  poimirlem15  38745  poimirlem18  38748  poimirlem21  38751  poimirlem22  38752  poimirlem24  38754  poimirlem26  38756  poimirlem27  38757  cdleme31sde  41634  cdlemeg47rv2  41759  dmmpossx2  49654  dfswapf2  50588  fucofvalg  50645  dfinito4  50841