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Theorem csbeq2gOLD 39288
 Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 4138. csbeq2gOLD 39288 is derived from the virtual deduction proof csbeq2gVD 39648. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3686 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbeq2gOLD (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Proof of Theorem csbeq2gOLD
StepHypRef Expression
1 spsbc 3600 . 2 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
2 sbceqg 4129 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
31, 2sylibd 229 1 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1629   = wceq 1631   ∈ wcel 2145  [wsbc 3587  ⦋csb 3682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-csb 3683 This theorem is referenced by:  csbsngVD  39649  csbxpgVD  39650  csbresgVD  39651  csbrngVD  39652  csbima12gALTVD  39653
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