Theorem sbcbidv 3457

Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Hypothesis

Ref	Expression
sbcbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbcbidv	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcbidv

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		sbcbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	sbcbid 3456	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 195  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403

This theorem is referenced by:  sbcbii  3458  opelopabsb  4910  opelopabgf  4920  opelopabf  4925  sbcfng  5955  sbcfg  5956  fmptsnd  6340  wrd2ind  13329  islmod  18690  elmptrab  21440  nbgraopALT  25953  f1od2  28887  isomnd  29032  isorng  29130  indexa  32698  sdclem2  32708  sdclem1  32709  fdc  32711  sbcalf  33087  sbcexf  33088  hdmap1ffval  36103  hdmap1fval  36104  hdmapffval  36136  hdmapfval  36137  hgmapffval  36195  hgmapfval  36196  rexrabdioph  36376  rexfrabdioph  36377  2rexfrabdioph  36378  3rexfrabdioph  36379  4rexfrabdioph  36380  6rexfrabdioph  36381  7rexfrabdioph  36382  2sbc6g  37638  2sbc5g  37639