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Theorem csbied2 3104
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
csbied2.1 (𝜑𝐴𝑉)
csbied2.2 (𝜑𝐴 = 𝐵)
csbied2.3 ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
csbied2 (𝜑𝐴 / 𝑥𝐶 = 𝐷)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbied2
StepHypRef Expression
1 csbied2.1 . 2 (𝜑𝐴𝑉)
2 id 19 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 csbied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2232 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 csbied2.3 . . 3 ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)
64, 5syldan 282 . 2 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
71, 6csbied 3103 1 (𝜑𝐴 / 𝑥𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by: (None)
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