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Theorem ovanraleqv 5798
 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 5782 . . . 4 (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋))
32eqeq1d 2148 . . 3 (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶))
41, 3anbi12d 464 . 2 (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
54ralbidv 2437 1 (𝐵 = 𝑋 → (∀𝑥𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∀wral 2416  (class class class)co 5774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777 This theorem is referenced by: (None)
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