Theorem sbceq1d 3796

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Hypothesis

Ref	Expression
sbceq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
sbceq1d	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))

Proof of Theorem sbceq1d

Step	Hyp	Ref	Expression
1		sbceq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		dfsbcq 3793	. 2 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814  df-sbc 3792

This theorem is referenced by:  sbceq1dd  3797  sbcnestgfw  4427  sbcnestgf  4432  ralrnmptw  7114  ralrnmpt  7116  tfindes  7884  findes  7923  frpoins3xpg  8164  frpoins3xp3g  8165  findcard2  9203  ac6sfi  9318  indexfi  9398  ac6num  10517  nn1suc  12286  uzind4s  12948  uzind4s2  12949  fzrevral  13649  fzshftral  13652  fi1uzind  14543  wrdind  14757  wrd2ind  14758  cjth  15139  prmind2  16719  isprs  18354  isdrs  18359  joinlem  18441  meetlem  18455  istos  18476  isdlat  18580  gsumvalx  18702  mndind  18854  issrg  20206  islmod  20879  quotval  26349  nn0min  32827  wrdt2ind  32923  bnj944  34931  sdclem2  37729  fdc  37732  hdmap1ffval  41778  hdmap1fval  41779  rexrabdioph  42782  2nn0ind  42934  zindbi  42935  iotasbcq  44433  prproropreud  47434