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Mirrors > Home > MPE Home > Th. List > ceqsex2v | Structured version Visualization version GIF version |
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Ref | Expression |
---|---|
ceqsex2v.1 | ⊢ 𝐴 ∈ V |
ceqsex2v.2 | ⊢ 𝐵 ∈ V |
ceqsex2v.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ceqsex2v.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ceqsex2v | ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 2013 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | nfv 2013 | . 2 ⊢ Ⅎ𝑦𝜒 | |
3 | ceqsex2v.1 | . 2 ⊢ 𝐴 ∈ V | |
4 | ceqsex2v.2 | . 2 ⊢ 𝐵 ∈ V | |
5 | ceqsex2v.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | ceqsex2v.4 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
7 | 1, 2, 3, 4, 5, 6 | ceqsex2 3461 | 1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1111 = wceq 1656 ∃wex 1878 ∈ wcel 2164 Vcvv 3414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-v 3416 |
This theorem is referenced by: ceqsex3v 3463 ceqsex4v 3464 ispos 17307 elfuns 32556 brimg 32578 brapply 32579 brsuccf 32582 brrestrict 32590 dfrdg4 32592 diblsmopel 37241 |
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