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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsaltv | Structured version Visualization version GIF version |
Description: Version of bj-ceqsalt 32069 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1868 and df-clab 2597. Prefer its use over bj-ceqsalt 32069 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ceqsaltv | ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-elissetv 32055 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | 1 | 3anim3i 1243 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴)) |
3 | bj-ceqsalt0 32067 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | |
4 | 2, 3 | syl 17 | 1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-ex 1696 df-nf 1701 df-clel 2606 |
This theorem is referenced by: bj-ceqsalgvALT 32075 |
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