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Theorem eqsbc3r 3835
Description: eqsbc3 3815 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 3815 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
2 eqcom 2826 . . 3 (𝐵 = 𝑥𝑥 = 𝐵)
32sbcbii 3827 . 2 ([𝐴 / 𝑥]𝐵 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐵)
4 eqcom 2826 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
51, 3, 43bitr4g 316 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1531  wcel 2108  [wsbc 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-sbc 3771
This theorem is referenced by:  sbcoreleleq  40859  sbcoreleleqVD  41183
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