Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ceqsal | 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.) |
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.2 | . 2 ⊢ 𝐴 ∈ V | |
2 | ceqsal.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | ceqsal.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | ceqsalg 2582 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: ceqsalv 2584 |
Copyright terms: Public domain | W3C validator |