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Theorem csbiebg 3522
 Description: Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
csbiebg.2 𝑥𝐶
Assertion
Ref Expression
csbiebg (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebg
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2621 . . . 4 (𝑎 = 𝐴 → (𝑥 = 𝑎𝑥 = 𝐴))
21imbi1d 330 . . 3 (𝑎 = 𝐴 → ((𝑥 = 𝑎𝐵 = 𝐶) ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
32albidv 1836 . 2 (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
4 csbeq1 3502 . . 3 (𝑎 = 𝐴𝑎 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
54eqeq1d 2612 . 2 (𝑎 = 𝐴 → (𝑎 / 𝑥𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
6 vex 3176 . . 3 𝑎 ∈ V
7 csbiebg.2 . . 3 𝑥𝐶
86, 7csbieb 3521 . 2 (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ 𝑎 / 𝑥𝐵 = 𝐶)
93, 5, 8vtoclbg 3240 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500 This theorem is referenced by:  cdlemefrs29bpre0  34702
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