Theorem raleqtrdv 2698

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 2696	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
4	1, 3	mpbid 147	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   = wceq 1364  ∀wral 2472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477

This theorem is referenced by:  znf1o  14139