Theorem ovanraleqv 6902

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 6886	. . . 4 ⊢ (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋))
3	2	eqeq1d 2801	. . 3 ⊢ (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶))
4	1, 3	anbi12d 625	. 2 ⊢ (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
5	4	ralbidv 3167	1 ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653  ∀wral 3089  (class class class)co 6878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881

This theorem is referenced by:  mgmidmo  17574  ismgmid  17579  ismgmid2  17582  mgmidsssn0  17584  gsumvalx  17585  gsumress  17591  ismndd  17628  mnd1  17646  gsumvallem2  17687  mhmmnd  17853  rngurd  30304  signsw0g  31151  signswmnd  31152  exidu1  34142  cmpidelt  34145  exidres  34164  exidresid  34165  isrngod  34184  rngoideu  34189  zlidlring  42727  2zrngamnd  42740