![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 474 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rexbida 3232 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 Ⅎwnf 1827 ∈ wcel 2107 ∃wrex 3091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-12 2163 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 df-nf 1828 df-rex 3096 |
This theorem is referenced by: rexbidvALT 3238 rexeqbid 3347 scott0 9046 infcvgaux1i 14993 bnj1463 31722 poimirlem25 34060 poimirlem26 34061 elrnmptf 40290 smfsupmpt 41948 smfinfmpt 41952 |
Copyright terms: Public domain | W3C validator |