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Mirrors > Home > MPE Home > Th. List > rexcom4b | Structured version Visualization version GIF version |
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Ref | Expression |
---|---|
rexcom4b.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rexcom4b | ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4a 3226 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | rexcom4b.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
3 | 2 | isseti 3209 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 526 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 3041 | . 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 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 |
This theorem is referenced by: islshpat 34304 |
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