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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 3171. (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 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rexbida 3251 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1784 ∈ wcel 2105 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1781 df-nf 1785 df-rex 3071 |
This theorem is referenced by: rexbidvALT 3254 rexeqbid 3327 scott0 9743 infcvgaux1i 15668 bnj1463 33334 fvineqsneq 35688 poimirlem25 35907 poimirlem26 35908 elrnmptf 43045 smfsupmpt 44690 smfinfmpt 44694 |
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