Theorem ovanraleqv 7380

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 7364	. . . 4 ⊢ (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋))
3	2	eqeq1d 2737	. . 3 ⊢ (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶))
4	1, 3	anbi12d 633	. 2 ⊢ (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
5	4	ralbidv 3158	1 ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∀wral 3049  (class class class)co 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6443  df-fv 6495  df-ov 7359

This theorem is referenced by:  mgmidmo  18617  ismgmid  18622  ismgmid2  18625  mgmidsssn0  18629  gsumvalx  18633  gsumress  18639  sgrpidmnd  18696  ismndd  18713  mnd1  18736  gsumvallem2  18791  mhmmnd  19029  ringurd  20155  opprring  20316  pzriprnglem7  21456  pzriprnglem13  21462  zsoring  28389  signsw0g  34688  signswmnd  34689  exidu1  38165  cmpidelt  38168  exidres  38187  exidresid  38188  isrngod  38207  rngoideu  38212  zlidlring  48698  2zrngamnd  48711