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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 2713. (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 2209 | . 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 1816 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)) |
6 | ceqsal.2 | . . . 4 ⊢ 𝐴 ∈ V | |
7 | 6 | isseti 3496 | . . 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 1535 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-clel 2814 |
This theorem is referenced by: ceqsex 3528 ralxp3f 8161 aomclem6 43048 |
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