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Mirrors > Home > MPE Home > Th. List > ceqsalgALT | Structured version Visualization version GIF version |
Description: Alternate proof of ceqsalg 3527, not using ceqsalt 3525. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsalgALT | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 3503 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | nfa1 2146 | . . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑) | |
3 | ceqsalg.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | ceqsalg.2 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 230 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 5 | a2i 14 | . . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
7 | 6 | sps 2174 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓)) |
8 | 2, 3, 7 | exlimd 2208 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
9 | 1, 8 | syl5com 31 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)) |
10 | 4 | biimprcd 251 | . . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 3, 10 | alrimi 2203 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
12 | 9, 11 | impbid1 226 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: (None) |
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