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Mirrors > Home > ILE Home > Th. List > rexbid | GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.) |
Ref | Expression |
---|---|
ralbid.1 | ⊢ Ⅎ𝑥𝜑 |
ralbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexbid | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbid.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | ralbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | adantr 261 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rexbida 2321 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1349 ∈ wcel 1393 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-rex 2312 |
This theorem is referenced by: rexbidv 2327 sbcrext 2835 caucvgsrlemgt1 6879 sscoll2 10113 |
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