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Theorem frege55lem1c 37727
 Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
Assertion
Ref Expression
frege55lem1c ((𝜑[𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem frege55lem1c
StepHypRef Expression
1 df-sbc 3422 . . 3 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 ∈ {𝑥𝑥 = 𝐵})
2 eqeq1 2625 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
32elabg 3338 . . . 4 (𝐴 ∈ {𝑥𝑥 = 𝐵} → (𝐴 ∈ {𝑥𝑥 = 𝐵} ↔ 𝐴 = 𝐵))
43ibi 256 . . 3 (𝐴 ∈ {𝑥𝑥 = 𝐵} → 𝐴 = 𝐵)
51, 4sylbi 207 . 2 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵)
65imim2i 16 1 ((𝜑[𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  {cab 2607  [wsbc 3421 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 3191  df-sbc 3422 This theorem is referenced by:  frege56c  37730
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