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Theorem csbeq2gOLD 38244
 Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3965. csbeq2gOLD 38244 is derived from the virtual deduction proof csbeq2gVD 38608. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3518 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 3430 . 2 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
2 sbceqg 3956 . 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 1478   = wceq 1480   ∈ wcel 1987  [wsbc 3417  ⦋csb 3514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-sbc 3418  df-csb 3515 This theorem is referenced by:  csbsngVD  38609  csbxpgVD  38610  csbresgVD  38611  csbrngVD  38612  csbima12gALTVD  38613
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