Theorem sbcied 3800

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2142, ax-12 2178. (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 3757	. 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 3640	. 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 1540   ∈ wcel 2109  {cab 2708  [wsbc 3756

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-sbc 3757

This theorem is referenced by:  sbcied2  3801  sbc2ie  3832  sbc2iedv  3833  sbc3ie  3834  sbcralt  3838  csbied  3901  euotd  5476  fmptsnd  7146  riota5f  7375  fpwwe2lem11  10601  fpwwe2lem12  10602  brfi1uzind  14480  opfi1uzind  14483  sbcie3s  17139  issubc  17804  gsumvalx  18610  dmdprd  19937  dprdval  19942  issrg  20104  issrng  20760  islmhm  20941  isphl  21544  istmd  23968  istgp  23971  isnlm  24570  isclm  24971  iscph  25077  iscms  25252  limcfval  25780  ewlksfval  29536  sbcies  32424  abfmpeld  32585  abfmpel  32586  isomnd  33022  isorng  33284  rprmval  33494  f1o2d2  42228