Theorem ovanraleqv 7433

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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ovanraleqv	⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (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

Step	Hyp	Ref	Expression
1		ovanraleqv.1	. . 3 ⊢ (𝐵 = 𝑋 → (𝜑 ↔ 𝜓))
2		oveq2 7417	. . . 4 ⊢ (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋))
3	2	eqeq1d 2762	. . 3 ⊢ (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶))
4	1, 3	anbi12d 644	. 2 ⊢ (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
5	4	ralbidv 3185	1 ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∀wral 3076  (class class class)co 7409

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6484  df-fv 6536  df-ov 7412

This theorem is used by:  mgmidmo  18785  ismgmid  18792  0gisid  18795  ismgmid2  18796  mgmidsssn0  18800  idressidex0  18807  gsumvalx  18812  gsumress  18818  sgrpidmnd  18875  ismndd  18893  mnd1  18920  gsumvallem2  18977  mhmmnd  19221  ringurd  20358  opprring  20524  pzriprnglem7  21740  pzriprnglem13  21746  zsoring  28714  signsw0g  35105  signswmnd  35106  exidu1  38704  cmpidelt  38707  exidres  38726  exidresid  38727  isrngod  38746  rngoideu  38751  zlidlring  49247  2zrngamnd  49260