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Theorem csbied2 2887
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
csbied2.1 (φA 𝑉)
csbied2.2 (φA = B)
csbied2.3 ((φ x = 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 (φA 𝑉)
2 id 19 . . . 4 (x = Ax = A)
3 csbied2.2 . . . 4 (φA = B)
42, 3sylan9eqr 2091 . . 3 ((φ x = A) → x = B)
5 csbied2.3 . . 3 ((φ x = B) → 𝐶 = 𝐷)
64, 5syldan 266 . 2 ((φ x = A) → 𝐶 = 𝐷)
71, 6csbied 2886 1 (φA / x𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  csb 2846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by: (None)
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