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| Mirrors > Home > MPE Home > Th. List > rexbid | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for restricted existential quantifier (deduction form). For a version based on fewer axioms see rexbidv 3178. (Contributed by NM, 27-Jun-1998.) | 
| Ref | Expression | 
|---|---|
| rexbid.1 | ⊢ Ⅎ𝑥𝜑 | 
| rexbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rexbid | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexbid.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | rexbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | 
| 4 | 1, 3 | rexbida 3271 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1782 ∈ wcel 2107 ∃wrex 3069 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-rex 3070 | 
| This theorem is referenced by: rexbidvALT 3274 rexeqbid 3356 scott0 9927 infcvgaux1i 15894 bnj1463 35070 fvineqsneq 37414 poimirlem25 37653 poimirlem26 37654 elrnmptf 45191 smfsupmpt 46835 smfinfmpt 46839 | 
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