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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf0 | GIF version |
Description: Lemma for elabgf 2902. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
elabgf0 | ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2256 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
2 | abid 2181 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
3 | 1, 2 | bitr3di 195 | 1 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 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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