Theorem cbvrmo 3424

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbvrmow 3404, cbvrmovw 3398 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

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

Proof of Theorem cbvrmo

Step	Hyp	Ref	Expression
1		cbvrmo.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvrmo.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvrmo.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvrex 3358	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
5	1, 2, 3	cbvreu 3423	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
6	4, 5	imbi12i 350	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓))
7		rmo5 3395	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
8		rmo5 3395	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓))
9	6, 7, 8	3bitr4i 303	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1784  ∃wrex 3069  ∃!wreu 3373  ∃*wrmo 3374

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376

This theorem is referenced by:  cbvrmov  3425