Theorem cbvrexfw 3303

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3346 with a disjoint variable condition, which does not require ax-13 2401. For a version not dependent on ax-11 2193 and ax-12, see cbvrexvw 3241. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2177, ax-13 2401. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvrexfw.1	⊢ Ⅎ𝑥𝐴
cbvrexfw.2	⊢ Ⅎ𝑦𝐴
cbvrexfw.3	⊢ Ⅎ𝑦𝜑
cbvrexfw.4	⊢ Ⅎ𝑥𝜓
cbvrexfw.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvrexfw

Step	Hyp	Ref	Expression
1		cbvrexfw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		cbvrexfw.2	. . . 4 ⊢ Ⅎ𝑦𝐴
3		cbvrexfw.3	. . . . 5 ⊢ Ⅎ𝑦𝜑
4	3	nfn 1889	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvrexfw.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1889	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbvrexfw.5	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	notbid 321	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
9	1, 2, 4, 6, 8	cbvralfw 3302	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
10		ralnex 3088	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
11		ralnex 3088	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐴 𝜓)
12	9, 10, 11	3bitr3i 304	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐴 𝜓)
13	12	con4bii 324	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  Ⅎwnf 1815  Ⅎwnfc 2907  ∀wral 3076  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087

This theorem is used by:  cbvrexw  3305  reusv2lem4  5363  reusv2  5365  nnwof  13048  cbviunf  33201  ac6sf2  33267  dfimafnf  33281  acunirnmpt2f  33306  aciunf1lem  33307  bnj1400  35515  phpreu  38636  poimirlem26  38673  indexa  38776  evth2f  46140  fvelrnbf  46143  evthf  46152  eliin2f  46227  stoweidlem34  47152  ovnlerp  47680