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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 2714. (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 2210 | . 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 1818 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)) | 
| 6 | ceqsal.2 | . . . 4 ⊢ 𝐴 ∈ V | |
| 7 | 6 | isseti 3497 | . . 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 206 ∀wal 1537 = wceq 1539 ∃wex 1778 Ⅎwnf 1782 ∈ wcel 2107 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-clel 2815 | 
| This theorem is referenced by: ceqsex 3529 ralxp3f 8163 aomclem6 43076 | 
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