Theorem sbceqbid 3752

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 2829	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
4	1, 3	eleq12d 2857	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}))
5		df-sbc 3746	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
6		df-sbc 3746	. 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})
7	4, 5, 6	3bitr4g 316	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1561   ∈ wcel 2143  {cab 2741  [wsbc 3745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-sbc 3746

This theorem is referenced by:  sbcbidv  3800  frpoins3xpg  8120  frpoins3xp3g  8121  fpwwe2cbv  10599  fpwwe2lem2  10601  fpwwe2lem3  10602  fi1uzind  14530  isprs  18338  isdrs  18343  istos  18458  isdlat  18564  issrg  20248  islmod  20938  fdc  38249  hdmap1ffval  42424  hdmap1fval  42425  hdmapffval  42455  hdmapfval  42456  hgmapffval  42514  hgmapfval  42515  sbccomieg  43375  rexrabdioph  43376