Theorem cbvrexw 3305

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 3303 with more disjoint variable conditions. (Contributed by NM, 31-Jul-2003.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

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

Proof of Theorem cbvrexw

Step	Hyp	Ref	Expression
1		nfcv 2903	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2903	. 2 ⊢ Ⅎ𝑦𝐴
3		cbvralw.1	. 2 ⊢ Ⅎ𝑦𝜑
4		cbvralw.2	. 2 ⊢ Ⅎ𝑥𝜓
5		cbvralw.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvrexfw 3303	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1780  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069

This theorem is referenced by:  cbvrexsvw  3316  cbvrexsvwOLD  3319  cbvreuw  3408  reu8nf  3886  cbviun  5041  isarep1  6657  isarep1OLD  6658  fvelimad  6976  dffo3f  7126  elabrex  7262  elabrexg  7263  onminex  7822  boxcutc  8980  indexfi  9398  wdom2d  9618  hsmexlem2  10465  fprodle  16029  iundisj  25597  mbfsup  25713  iundisjf  32609  iundisjfi  32804  voliune  34210  volfiniune  34211  bnj1542  34850  cvmcov  35248  poimirlem24  37631  poimirlem26  37633  indexa  37720  mndmolinv  42077  primrootsunit1  42079  primrootsunit  42080  primrootspoweq0  42088  aks6d1c4  42106  aks6d1c6isolem1  42156  aks6d1c6isolem2  42157  rhmqusspan  42167  grpods  42176  unitscyglem1  42177  unitscyglem3  42179  unitscyglem4  42180  rexrabdioph  42782  rexfrabdioph  42783  disjrnmpt2  45131  caucvgbf  45440  limsuppnfd  45658  limsuppnf  45667  limsupre2  45681  limsupre3  45689  limsupre3uz  45692  limsupreuz  45693  liminfreuz  45759  stoweidlem31  45987  stoweidlem59  46015  rexsb  47049  cbvrex2  47054  2reu8i  47063