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Mirrors > Home > MPE Home > Th. List > rexxfr2d | Structured version Visualization version GIF version |
Description: Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
ralxfr2d.1 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) |
ralxfr2d.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) |
ralxfr2d.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexxfr2d | ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralxfr2d.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) | |
2 | ralxfr2d.2 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) | |
3 | ralxfr2d.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 3 | notbid 318 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒)) |
5 | 1, 2, 4 | ralxfr2d 5416 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐶 ¬ 𝜒)) |
6 | 5 | notbid 318 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝜒)) |
7 | dfrex2 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓) | |
8 | dfrex2 3071 | . 2 ⊢ (∃𝑦 ∈ 𝐶 𝜒 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝜒) | |
9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 |
This theorem is referenced by: rexrn 7107 reximaOLD 7259 cnpresti 23312 cnprest 23313 1stcrest 23477 subislly 23505 txrest 23655 trfil2 23911 met1stc 24550 metucn 24600 xrlimcnp 27026 esumlub 34041 esumfsup 34051 rexxfr3d 35623 ptrest 37606 djhcvat42 41398 |
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