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Mirrors > Home > ILE Home > Th. List > ceqsalg | GIF version |
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsalg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2568 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | nfa1 1434 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑) | |
3 | ceqsalg.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | ceqsalg.2 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 132 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 5 | a2i 11 | . . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
7 | 6 | sps 1430 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
8 | 2, 3, 7 | exlimd 1488 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
9 | 1, 8 | syl5com 26 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)) |
10 | 4 | biimprcd 149 | . . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 3, 10 | alrimi 1415 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
12 | 9, 11 | impbid1 130 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 ∈ wcel 1393 |
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: ceqsal 2583 sbc6g 2788 uniiunlem 3028 sucprcreg 4273 funimass4 5224 ralrnmpt2 5615 |
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