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Mirrors > Home > MPE Home > Th. List > rexcom4a | Structured version Visualization version GIF version |
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Ref | Expression |
---|---|
rexcom4a | ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3197 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) | |
2 | 19.42v 1912 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) | |
3 | 2 | rexbii 3195 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓)) |
4 | 1, 3 | bitr3i 269 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∃wex 1742 ∃wrex 3090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-11 2093 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-rex 3095 |
This theorem is referenced by: rexcom4b 3446 bj-rexcom4bv 33688 bj-rexcom4b 33689 |
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