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Mirrors > Home > MPE Home > Th. List > brimralrspcev | Structured version Visualization version GIF version |
Description: Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.) |
Ref | Expression |
---|---|
brimralrspcev | ⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5070 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
2 | 1 | anbi2d 630 | . 2 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝐴𝑅𝑥) ↔ (𝜑 ∧ 𝐴𝑅𝐵))) |
3 | 2 | rspceaimv 3628 | 1 ⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 |
This theorem is referenced by: dveflem 24576 mullimc 41946 limcdm0 41948 mullimcf 41953 constlimc 41954 idlimc 41956 limcleqr 41974 addlimc 41978 0ellimcdiv 41979 ioodvbdlimc1lem2 42266 ioodvbdlimc2lem 42268 |
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