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Theorem csbiedf 3910
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 413 . . 3 (𝜑 → (𝑥 = 𝐴𝐵 = 𝐶))
41, 3alrimi 2203 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
5 csbiedf.3 . . 3 (𝜑𝐴𝑉)
6 csbiedf.2 . . 3 (𝜑𝑥𝐶)
7 csbiebt 3909 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
85, 6, 7syl2anc 584 . 2 (𝜑 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
94, 8mpbid 233 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wnf 1775  wcel 2105  wnfc 2958  csb 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sbc 3770  df-csb 3881
This theorem is referenced by:  csbied  3916  csbie2t  3918  fprodsplit1f  15332  natpropd  17234  fucpropd  17235  gsummptf1o  19012  gsummpt2d  30614  sumsnd  41160  fsumsplit1  41729
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