![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2752 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | nfa1 1541 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑) | |
3 | ceqsalg.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | ceqsalg.2 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 144 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 5 | a2i 11 | . . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
7 | 6 | sps 1537 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
8 | 2, 3, 7 | exlimd 1597 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
9 | 1, 8 | syl5com 29 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)) |
10 | 4 | biimprcd 160 | . . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 3, 10 | alrimi 1522 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
12 | 9, 11 | impbid1 142 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 Ⅎwnf 1460 ∃wex 1492 ∈ wcel 2148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2740 |
This theorem is referenced by: ceqsal 2767 sbc6g 2988 uniiunlem 3245 sucprcreg 4549 funimass4 5567 ralrnmpo 5989 |
Copyright terms: Public domain | W3C validator |