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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsalg | Structured version Visualization version GIF version |
Description: Remove from ceqsalg 3527 dependency on ax-ext 2790 (and on df-cleq 2811 and df-v 3494). See also bj-ceqsalgv 34104. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
bj-ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-ceqsalg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-elisset 34089 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | bj-ceqsalg.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-ceqsalg.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | bj-ceqsalg0 34101 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-3an 1081 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-clel 2890 |
This theorem is referenced by: (None) |
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