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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsalg | Structured version Visualization version GIF version |
Description: Remove from ceqsalg 3516 dependency on ax-ext 2707 (and on df-cleq 2728 and df-v 3481). See also bj-ceqsalgv 36870. (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 | elisset 2822 | . 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 36867 | . 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 206 ∀wal 1538 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-12 2177 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-clel 2815 |
This theorem is referenced by: (None) |
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