Theorem cbvrmowOLD 3409

Description: Obsolete version of cbvrmow 3403 as of 23-May-2024. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvrmowOLD.1	⊢ Ⅎ𝑦𝜑
cbvrmowOLD.2	⊢ Ⅎ𝑥𝜓
cbvrmowOLD.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrmowOLD	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦,𝐴

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

Proof of Theorem cbvrmowOLD

Step	Hyp	Ref	Expression
1		cbvrmowOLD.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvrmowOLD.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvrmowOLD.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvrexw 3302	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
5	1, 2, 3	cbvreuw 3404	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
6	4, 5	imbi12i 349	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓))
7		rmo5 3394	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
8		rmo5 3394	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓))
9	6, 7, 8	3bitr4i 302	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1783  ∃wrex 3068  ∃!wreu 3372  ∃*wrmo 3373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-11 2152  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-mo 2532  df-eu 2561  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375

This theorem is referenced by: (None)