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Theorem elabgf0 11020
Description: Lemma for elabgf 2746. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
elabgf0 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))

Proof of Theorem elabgf0
StepHypRef Expression
1 abid 2071 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 eleq1 2145 . 2 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
31, 2syl5rbbr 193 1 (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  {cab 2069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079
This theorem is referenced by:  elabgft1  11021  elabgf2  11023
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