Theorem sbceqbid 3757

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 2795	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
4	1, 3	eleq12d 2822	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}))
5		df-sbc 3751	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
6		df-sbc 3751	. 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707  [wsbc 3750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sbc 3751

This theorem is referenced by:  sbcbidv  3806  frpoins3xpg  8096  frpoins3xp3g  8097  fpwwe2cbv  10559  fpwwe2lem2  10561  fpwwe2lem3  10562  fi1uzind  14448  isprs  18237  isdrs  18242  istos  18357  isdlat  18463  issrg  20108  islmod  20802  fdc  37732  hdmap1ffval  41782  hdmap1fval  41783  hdmapffval  41813  hdmapfval  41814  hgmapffval  41872  hgmapfval  41873  sbccomieg  42774  rexrabdioph  42775