Theorem rexlimdvaacbv 45346

Description: Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3168. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
rexlimdvaacbv.1	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
rexlimdvaacbv.2	⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)

Assertion

Ref	Expression
rexlimdvaacbv	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜓,𝑦   𝜃,𝑥   𝜑,𝑦   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝜃(𝑦)

Proof of Theorem rexlimdvaacbv

Step	Hyp	Ref	Expression
1		rexlimdvaacbv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
2	1	cbvrexv 3354	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜃)
3		rexlimdvaacbv.2	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)
4	3	rexlimdvaa 3168	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜃 → 𝜒))
5	2, 4	biimtrid 245	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-10 2179  ax-11 2196  ax-12 2215  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2101  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091

This theorem is used by:  rexlimddvcbv  45348