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Mirrors > Home > MPE Home > Th. List > ceqsexOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ceqsex 3523 as of 22-Jan-2025. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ceqsex.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsex.2 | ⊢ 𝐴 ∈ V |
ceqsex.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsexOLD | ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsex.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsex.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpa 478 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 2211 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
5 | 2 | biimprcd 249 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
6 | 1, 5 | alrimi 2207 | . . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
7 | ceqsex.2 | . . . 4 ⊢ 𝐴 ∈ V | |
8 | 7 | isseti 3490 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 |
9 | exintr 1896 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
10 | 6, 8, 9 | mpisyl 21 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
11 | 4, 10 | impbii 208 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 Ⅎwnf 1786 ∈ wcel 2107 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-clel 2811 |
This theorem is referenced by: (None) |
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