Theorem rexlimdvaacbv 44781

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 3164. (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 3352	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜃)
3		rexlimdvaacbv.2	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)
4	3	rexlimdvaa 3164	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜃 → 𝜒))
5	2, 4	biimtrid 244	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  ∃wrex 3086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087

This theorem is referenced by:  rexlimddvcbv  44783