![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsalg | Structured version Visualization version GIF version |
Description: Remove from ceqsalg 3502 dependency on ax-ext 2697 (and on df-cleq 2718 and df-v 3470). See also bj-ceqsalgv 36278. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
bj-ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-ceqsalg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2809 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | bj-ceqsalg.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-ceqsalg.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | bj-ceqsalg0 36275 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀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 |
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-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-clel 2804 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |