Theorem rexlimdvaacbv 44649

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 3141. (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 3329	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜃)
3		rexlimdvaacbv.2	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)
4	3	rexlimdvaa 3141	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜃 → 𝜒))
5	2, 4	biimtrid 243	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∃wrex 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064

This theorem is referenced by:  rexlimddvcbv  44651