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Theorem eqsbc3r 3625
Description: eqsbc3 3608 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
Assertion
Ref Expression
eqsbc3r (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqsbc3 3608 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
2 eqcom 2759 . . 3 (𝐵 = 𝑥𝑥 = 𝐵)
32sbcbii 3624 . 2 ([𝐴 / 𝑥]𝐵 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐵)
4 eqcom 2759 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
51, 3, 43bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1624  wcel 2131  [wsbc 3568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-v 3334  df-sbc 3569
This theorem is referenced by:  sbcoreleleq  39239  sbcoreleleqVD  39586
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