Theorem rexeqtrdv 3313

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

Hypotheses

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

Assertion

Ref	Expression
rexeqtrdv	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem rexeqtrdv

Step	Hyp	Ref	Expression
1		rexeqtrdv.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		rexeqtrdv.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	rexeqdv 3311	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   = wceq 1539  ∃wrex 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2726  df-rex 3060

This theorem is referenced by:  ballotlemfc0  34436  ballotlemfcc  34437  lkrlspeqN  39113