Theorem cbvrmo 3347

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 2371. Use the weaker cbvrmow 3342, cbvrmovw 3350 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvral.1	⊢ Ⅎ𝑦𝜑
cbvral.2	⊢ Ⅎ𝑥𝜓
cbvral.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		cbvral.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvral.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvral.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvrex 3345	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
5	1, 2, 3	cbvreu 3346	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
6	4, 5	imbi12i 354	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓))
7		rmo5 3331	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
8		rmo5 3331	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1791  ∃wrex 3052  ∃!wreu 3053  ∃*wrmo 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059

This theorem is referenced by:  cbvrmov  3356