Theorem raleqtrdv 3324

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 3322	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-ral 3079  df-rex 3089

This theorem is used by:  subgpgp  19673  znf1o  21712  perfopn  23353  ordtt1  23547  tx1stc  23818  xkococnlem  23827  dgrlem  26397  dchrisum0flb  27685  wlknewwlksn  30247  gsummoncoe1fz  33897  sigaclcu3  34521  subfacp1lem3  35682  poimirlem1  38300