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Mirrors > Home > MPE Home > Th. List > rmobidv | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmobidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rmobidv | ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmobidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
3 | 2 | rmobidva 3327 | 1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ∃*wrmo 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-mo 2540 df-rmo 3071 |
This theorem is referenced by: rmoeqd 3351 brdom7disj 10287 ddemeas 32204 poimirlem26 35803 |
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