Theorem rexeqtrrdv 3329

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

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem rexeqtrrdv

Step	Hyp	Ref	Expression
1		rexeqtrrdv.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		rexeqtrrdv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
3	2	rexeqdv 3325	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
4	1, 3	mpbird 257	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   = wceq 1537  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-rex 3069

This theorem is referenced by:  zornn0g  10543  ablfacrplem  20100  ablfac2  20124  2ndcctbss  23479  1stcelcls  23485  wwlksnextsurj  29930  primrootsunit1  42079