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Mirrors > Home > MPE Home > Th. List > ceqsal | Structured version Visualization version GIF version |
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid df-clab 2706. (Revised by Wolf Lammen, 23-Jan-2025.) |
Ref | Expression |
---|---|
ceqsal.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsal.2 | ⊢ 𝐴 ∈ V |
ceqsal.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsal | ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsal.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.23 2200 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
3 | ceqsal.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | pm5.74i 271 | . . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)) |
5 | 4 | albii 1814 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)) |
6 | ceqsal.2 | . . . 4 ⊢ 𝐴 ∈ V | |
7 | 6 | isseti 3487 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 |
8 | 7 | a1bi 362 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
9 | 2, 5, 8 | 3bitr4i 303 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 = wceq 1534 ∃wex 1774 Ⅎwnf 1778 ∈ wcel 2099 Vcvv 3471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 df-clel 2806 |
This theorem is referenced by: ceqsalvOLD 3510 ceqsex 3521 ralxp3f 8142 aomclem6 42483 |
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