Theorem sbceqbid 3733

Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)

Hypotheses

Ref	Expression
sbceqbid.1	⊢ (𝜑 → 𝐴 = 𝐵)
sbceqbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem sbceqbid

Step	Hyp	Ref	Expression
1		sbceqbid.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		sbceqbid.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	abbidv 2802	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
4	1, 3	eleq12d 2830	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}))
5		df-sbc 3727	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
6		df-sbc 3727	. 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})
7	4, 5, 6	3bitr4g 315	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1543   ∈ wcel 2115  {cab 2714  [wsbc 3726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3727

This theorem is referenced by:  sbcbidv  3781  frpoins3xpg  8083  frpoins3xp3g  8084  fpwwe2cbv  10547  fpwwe2lem2  10549  fpwwe2lem3  10550  fi1uzind  14463  isprs  18256  isdrs  18261  istos  18376  isdlat  18482  issrg  20163  islmod  20857  fdc  38109  hdmap1ffval  42284  hdmap1fval  42285  hdmapffval  42315  hdmapfval  42316  hgmapffval  42374  hgmapfval  42375  sbccomieg  43235  rexrabdioph  43236