Theorem raleqtrdv 3300

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 3298	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
4	1, 3	mpbid 233	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   = wceq 1547  ∀wral 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732  df-ral 3055  df-rex 3065

This theorem is referenced by:  subgpgp  19570  znf1o  21533  perfopn  23175  ordtt1  23369  tx1stc  23640  xkococnlem  23649  dgrlem  26219  dchrisum0flb  27498  wlknewwlksn  29980  gsummoncoe1fz  33688  sigaclcu3  34313  subfacp1lem3  35411  poimirlem1  37989