Theorem sbceq1d 3741

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 3738	. 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 1541  [wsbc 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2806  df-sbc 3737

This theorem is referenced by:  sbceq1dd  3742  sbcnestgfw  4366  sbcnestgf  4371  ralrnmptw  7022  ralrnmpt  7024  tfindes  7788  findes  7825  frpoins3xpg  8065  frpoins3xp3g  8066  findcard2  9069  ac6sfi  9163  indexfi  9239  ac6num  10365  nn1suc  12142  uzind4s  12801  uzind4s2  12802  fzrevral  13507  fzshftral  13510  fi1uzind  14409  wrdind  14624  wrd2ind  14625  cjth  15005  prmind2  16591  isprs  18197  isdrs  18202  joinlem  18282  meetlem  18296  istos  18317  isdlat  18423  gsumvalx  18579  mndind  18731  issrg  20101  islmod  20792  quotval  26222  nn0min  32795  wrdt2ind  32926  bnj944  34942  sdclem2  37782  fdc  37785  hdmap1ffval  41834  hdmap1fval  41835  rexrabdioph  42827  2nn0ind  42978  zindbi  42979  iotasbcq  44469  prproropreud  47540