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Theorem csbiedf 3818
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1 𝑥𝜑
csbiedf.2 (𝜑𝑥𝐶)
csbiedf.3 (𝜑𝐴𝑉)
csbiedf.4 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
csbiedf (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3 𝑥𝜑
2 csbiedf.4 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
32ex 416 . . 3 (𝜑 → (𝑥 = 𝐴𝐵 = 𝐶))
41, 3alrimi 2214 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
5 csbiedf.3 . . 3 (𝜑𝐴𝑉)
6 csbiedf.2 . . 3 (𝜑𝑥𝐶)
7 csbiebt 3817 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
85, 6, 7syl2anc 587 . 2 (𝜑 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
94, 8mpbid 235 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  wnf 1790  wcel 2113  wnfc 2879  csb 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3399  df-sbc 3680  df-csb 3789
This theorem is referenced by:  csbied  3824  csbie2t  3826  fvmptdf  6775  fprodsplit1f  15429  natpropd  17344  fucpropd  17345  gsummptf1o  19195  gsummpt2d  30878  mnringvald  41357  sumsnd  42091  fsumsplit1  42639
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