![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2704. (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 2196 | . 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 1813 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)) |
6 | ceqsal.2 | . . . 4 ⊢ 𝐴 ∈ V | |
7 | 6 | isseti 3484 | . . 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 1531 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 ∈ wcel 2098 Vcvv 3468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-clel 2804 |
This theorem is referenced by: ceqsalvOLD 3507 ceqsex 3518 ralxp3f 8120 aomclem6 42360 |
Copyright terms: Public domain | W3C validator |