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Theorem csbiedf 3089
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 114 . . 3 (𝜑 → (𝑥 = 𝐴𝐵 = 𝐶))
41, 3alrimi 1515 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
5 csbiedf.3 . . 3 (𝜑𝐴𝑉)
6 csbiedf.2 . . 3 (𝜑𝑥𝐶)
7 csbiebt 3088 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
85, 6, 7syl2anc 409 . 2 (𝜑 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
94, 8mpbid 146 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346   = wceq 1348  wnf 1453  wcel 2141  wnfc 2299  csb 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050
This theorem is referenced by:  csbied  3095  csbie2t  3097  fprodsplit1f  11590
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