Theorem raleqtrdv 3316

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)

Hypotheses

Ref	Expression
raleqtrdv.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
raleqtrdv.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
raleqtrdv	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem raleqtrdv

Step	Hyp	Ref	Expression
1		raleqtrdv.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
2		raleqtrdv.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	raleqdv 3314	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
4	1, 3	mpbid 234	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   = wceq 1554  ∀wral 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794  df-cleq 2748  df-ral 3071  df-rex 3081

This theorem is referenced by:  subgpgp  19613  znf1o  21576  perfopn  23218  ordtt1  23412  tx1stc  23683  xkococnlem  23692  dgrlem  26262  dchrisum0flb  27544  wlknewwlksn  30026  gsummoncoe1fz  33748  sigaclcu3  34373  subfacp1lem3  35480  poimirlem1  38068