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Theorem frege55lem1c 38737
 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 3589 . . 3 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 ∈ {𝑥𝑥 = 𝐵})
2 eqeq1 2775 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
32elabg 3503 . . . 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 1631   ∈ wcel 2145  {cab 2757  [wsbc 3588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3589 This theorem is referenced by:  frege56c  38740
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