Theorem rexxfr3d 35680

Description: Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by SN, 20-Jun-2025.)

Hypotheses

Ref	Expression
rexxfr3d.s	⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))
rexxfr3d.x	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))
rexxfr3d.a	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
rexxfr3d	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥   𝑥,𝑋   𝑥,𝐵   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem rexxfr3d

Step	Hyp	Ref	Expression
1		rexxfr3d.a	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2	1	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝑉)
3		rexxfr3d.x	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))
4		rexxfr3d.s	. . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))
5	4	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝜓 ↔ 𝜒))
6	2, 3, 5	rexxfr2d 5349	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057

This theorem is referenced by:  ellcsrspsn  35683