Theorem cbvrexfw 3303

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3358 with a disjoint variable condition, which does not require ax-13 2372. For a version not dependent on ax-11 2155 and ax-12, see cbvrexvw 3236. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2138, ax-13 2372. (Revised by Gino Giotto, 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 1861	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvrexfw.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1861	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbvrexfw.5	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	notbid 318	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
9	1, 2, 4, 6, 8	cbvralfw 3302	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
10		ralnex 3073	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
11		ralnex 3073	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐴 𝜓)
12	9, 10, 11	3bitr3i 301	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐴 𝜓)
13	12	con4bii 321	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  Ⅎwnf 1786  Ⅎwnfc 2884  ∀wral 3062  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072

This theorem is referenced by:  cbvrexw  3305  reusv2lem4  5400  reusv2  5402  nnwof  12898  cbviunf  31787  ac6sf2  31849  dfimafnf  31860  aciunf1lem  31887  bnj1400  33846  phpreu  36472  poimirlem26  36514  indexa  36601  evth2f  43699  fvelrnbf  43702  evthf  43711  eliin2f  43793  stoweidlem34  44750  ovnlerp  45278