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Theorem ovanraleqv 7157
Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
Hypothesis
Ref Expression
ovanraleqv.1 (𝐵 = 𝑋 → (𝜑𝜓))
Assertion
Ref Expression
ovanraleqv (𝐵 = 𝑋 → (∀𝑥𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝐶(𝑥)   · (𝑥)   𝑉(𝑥)

Proof of Theorem ovanraleqv
StepHypRef Expression
1 ovanraleqv.1 . . 3 (𝐵 = 𝑋 → (𝜑𝜓))
2 oveq2 7141 . . . 4 (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋))
32eqeq1d 2822 . . 3 (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶))
41, 3anbi12d 632 . 2 (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
54ralbidv 3184 1 (𝐵 = 𝑋 → (∀𝑥𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wral 3125  (class class class)co 7133
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 2792
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 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-iota 6290  df-fv 6339  df-ov 7136
This theorem is referenced by:  mgmidmo  17849  ismgmid  17854  ismgmid2  17857  mgmidsssn0  17861  gsumvalx  17865  gsumress  17871  sgrpidmnd  17895  ismndd  17912  mnd1  17931  gsumvallem2  17977  mhmmnd  18200  rngurd  30865  signsw0g  31834  signswmnd  31835  exidu1  35170  cmpidelt  35173  exidres  35192  exidresid  35193  isrngod  35212  rngoideu  35217  zlidlring  44344  2zrngamnd  44357
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