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Mirrors > Home > MPE Home > Th. List > ceqsrexbv | Structured version Visualization version GIF version |
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |
Ref | Expression |
---|---|
ceqsrexv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsrexbv | ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 3073 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑))) | |
2 | eleq1 2676 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 3 | pm5.32ri 668 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑))) |
5 | 4 | bicomi 213 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑))) |
6 | 5 | baib 942 | . . 3 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ 𝜑))) |
7 | 6 | rexbiia 3022 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑)) |
8 | ceqsrexv.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
9 | 8 | ceqsrexv 3306 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
10 | 9 | pm5.32i 667 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
11 | 1, 7, 10 | 3bitr3i 289 | 1 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-rex 2902 df-v 3175 |
This theorem is referenced by: marypha2lem2 8225 txkgen 21265 ceqsrexv2 30860 eq0rabdioph 36358 |
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