Theorem ovanraleqv 7383

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 7367	. . . 4 ⊢ (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋))
3	2	eqeq1d 2743	. . 3 ⊢ (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶))
4	1, 3	anbi12d 639	. 2 ⊢ (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
5	4	ralbidv 3164	1 ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∀wral 3055  (class class class)co 7359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362

This theorem is referenced by:  mgmidmo  18623  ismgmid  18628  ismgmid2  18631  mgmidsssn0  18635  gsumvalx  18639  gsumress  18645  sgrpidmnd  18702  ismndd  18719  mnd1  18742  gsumvallem2  18797  mhmmnd  19035  ringurd  20160  opprring  20321  pzriprnglem7  21465  pzriprnglem13  21471  zsoring  28421  signsw0g  34750  signswmnd  34751  exidu1  38236  cmpidelt  38239  exidres  38258  exidresid  38259  isrngod  38278  rngoideu  38283  zlidlring  48737  2zrngamnd  48750