Theorem csbeq2dv 3167

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

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜑
2		csbeq2dv.1	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
3	1, 2	csbeq2d 3166	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1398  ⦋csb 3141

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3046  df-csb 3142

This theorem is referenced by:  csbeq2i  3168  mpomptsx  6406  dmmpossx  6408  fmpox  6409  fmpoco  6425  fisumcom2  12152  fprodcom2fi  12340  imasex  13572  prdsex  14117  fsumcncntop  15561  dvmptfsum  15719