Theorem rexeqtrdv 3329

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 3327	. 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 1570  ∃wrex 3092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2758  df-rex 3093

This theorem is used by:  dflringlem  33815  ballotlemfc0  34915  ballotlemfcc  34916  lkrlspeqN  39986