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Theorem csbiedf 3928
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 412 . . 3 (𝜑 → (𝑥 = 𝐴𝐵 = 𝐶))
41, 3alrimi 2212 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
5 csbiedf.3 . . 3 (𝜑𝐴𝑉)
6 csbiedf.2 . . 3 (𝜑𝑥𝐶)
7 csbiebt 3927 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
85, 6, 7syl2anc 584 . 2 (𝜑 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
94, 8mpbid 232 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wnf 1782  wcel 2107  wnfc 2889  csb 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481  df-sbc 3788  df-csb 3899
This theorem is referenced by:  csbie2t  3936  fvmptdf  7021  fsumsplit1  15782  fprodsplit1f  16027  natpropd  18025  fucpropd  18026  gsummptf1o  19982  gsummpt2d  33053  gsummptfsf1o  33058  mnringvald  44232  sumsnd  45036
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