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Theorem elabgf0 15214
Description: Lemma for elabgf 2902. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
elabgf0 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))

Proof of Theorem elabgf0
StepHypRef Expression
1 eleq1 2256 . 2 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
2 abid 2181 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
31, 2bitr3di 195 1 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2164  {cab 2179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189
This theorem is referenced by:  elabgft1  15215  elabgf2  15217
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