Theorem cbvrexfw 3277

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3331 with a disjoint variable condition, which does not require ax-13 2376. For a version not dependent on ax-11 2162 and ax-12, see cbvrexvw 3215. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2146, ax-13 2376. (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 1858	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
5		cbvrexfw.4	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1858	. . . 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 3276	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
10		ralnex 3062	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
11		ralnex 3062	. . 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 206  Ⅎwnf 1784  Ⅎwnfc 2883  ∀wral 3051  ∃wrex 3060

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-11 2162  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061

This theorem is referenced by:  cbvrexw  3279  reusv2lem4  5346  reusv2  5348  nnwof  12827  cbviunf  32630  ac6sf2  32700  dfimafnf  32714  aciunf1lem  32740  bnj1400  34991  phpreu  37805  poimirlem26  37847  indexa  37934  evth2f  45260  fvelrnbf  45263  evthf  45272  eliin2f  45348  stoweidlem34  46278  ovnlerp  46806