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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rexcom4b | Structured version Visualization version GIF version |
Description: Remove from rexcom4b 3258 dependency on ax-ext 2631 and ax-13 2282 (and on df-or 384, df-cleq 2644, df-nfc 2782, df-v 3233). The hypothesis uses 𝑉 instead of V (see bj-isseti 32989 for the motivation). Use bj-rexcom4bv 32996 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-rexcom4b.1 | ⊢ 𝐵 ∈ 𝑉 |
Ref | Expression |
---|---|
bj-rexcom4b | ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rexcom4a 32995 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | bj-rexcom4b.1 | . . . . 5 ⊢ 𝐵 ∈ 𝑉 | |
3 | 2 | bj-isseti 32989 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 525 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 3070 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
6 | 1, 5 | bitr4i 267 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 ∃wrex 2942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-11 2074 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1745 df-sb 1938 df-clab 2638 df-clel 2647 df-rex 2947 |
This theorem is referenced by: (None) |
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