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Theorem csbeq2gOLD 39082
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 4026. csbeq2gOLD 39082 is derived from the virtual deduction proof csbeq2gVD 39442. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3570 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 3481 . 2 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
2 sbceqg 4017 . 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 1521   = wceq 1523  wcel 2030  [wsbc 3468  csb 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567
This theorem is referenced by:  csbsngVD  39443  csbxpgVD  39444  csbresgVD  39445  csbrngVD  39446  csbima12gALTVD  39447
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