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Mirrors > Home > MPE Home > Th. List > rexxfr | Structured version Visualization version GIF version |
Description: Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
Ref | Expression |
---|---|
ralxfr.1 | ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) |
ralxfr.2 | ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) |
ralxfr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexxfr | ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 2996 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜑) | |
2 | dfrex2 2996 | . . 3 ⊢ (∃𝑦 ∈ 𝐶 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝜓) | |
3 | ralxfr.1 | . . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) | |
4 | ralxfr.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
5 | ralxfr.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 5 | notbid 308 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
7 | 3, 4, 6 | ralxfr 4886 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐶 ¬ 𝜓) |
8 | 2, 7 | xchbinxr 325 | . 2 ⊢ (∃𝑦 ∈ 𝐶 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜑) |
9 | 1, 8 | bitr4i 267 | 1 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 |
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: infm3 10982 reeff1o 24201 moxfr 37255 |
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