Theorem sbcied 3785

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2144, ax-12 2180. (Revised by GG, 12-Oct-2024.)

Hypotheses

Ref	Expression
sbcied.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sbcied.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied

Step	Hyp	Ref	Expression
1		df-sbc 3742	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		sbcied.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		sbcied.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	2, 3	elabd3 3626	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
5	1, 4	bitrid 283	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  [wsbc 3741

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-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3742

This theorem is referenced by:  sbcied2  3786  sbc2ie  3817  sbc2iedv  3818  sbc3ie  3819  sbcralt  3823  csbied  3886  euotd  5453  fmptsnd  7103  riota5f  7331  fpwwe2lem11  10532  fpwwe2lem12  10533  brfi1uzind  14415  opfi1uzind  14418  sbcie3s  17073  issubc  17742  gsumvalx  18584  dmdprd  19913  dprdval  19918  isomnd  20036  issrg  20107  issrng  20760  isorng  20777  islmhm  20962  isphl  21566  istmd  23990  istgp  23993  isnlm  24591  isclm  24992  iscph  25098  iscms  25273  limcfval  25801  ewlksfval  29581  sbcies  32465  abfmpeld  32634  abfmpel  32635  rprmval  33479  f1o2d2  42272