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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). (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 3315 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1775 ∈ wcel 2105 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-rex 3141 |
This theorem is referenced by: rexbidvALT 3318 rexeqbid 3420 scott0 9303 infcvgaux1i 15200 bnj1463 32224 fvineqsneq 34575 poimirlem25 34798 poimirlem26 34799 elrnmptf 41317 smfsupmpt 42966 smfinfmpt 42970 |
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