Theorem sbceqbid 3743

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 2797	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
4	1, 3	eleq12d 2825	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}))
5		df-sbc 3737	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
6		df-sbc 3737	. 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 1541   ∈ wcel 2111  {cab 2709  [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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737

This theorem is referenced by:  sbcbidv  3792  frpoins3xpg  8076  frpoins3xp3g  8077  fpwwe2cbv  10527  fpwwe2lem2  10529  fpwwe2lem3  10530  fi1uzind  14420  isprs  18208  isdrs  18213  istos  18328  isdlat  18434  issrg  20112  islmod  20803  fdc  37791  hdmap1ffval  41900  hdmap1fval  41901  hdmapffval  41931  hdmapfval  41932  hgmapffval  41990  hgmapfval  41991  sbccomieg  42891  rexrabdioph  42892