Theorem cbvrexw 3302

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 3300 with more disjoint variable conditions. (Contributed by NM, 31-Jul-2003.) Avoid ax-13 2369. (Revised by Gino Giotto, 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 2901	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2901	. 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 3300	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1783  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-11 2152  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069

This theorem is referenced by:  cbvrexsvw  3313  cbvrexsvwOLD  3315  cbvreuw  3404  cbvrmowOLD  3409  reu8nf  3870  cbviun  5038  isarep1  6636  isarep1OLD  6637  fvelimad  6958  dffo3f  7106  elabrex  7243  elabrexg  7244  onminex  7792  boxcutc  8937  indexfi  9362  wdom2d  9577  hsmexlem2  10424  fprodle  15944  iundisj  25297  mbfsup  25413  iundisjf  32087  iundisjfi  32274  voliune  33525  volfiniune  33526  bnj1542  34166  cvmcov  34552  poimirlem24  36815  poimirlem26  36817  indexa  36904  rexrabdioph  41834  rexfrabdioph  41835  disjrnmpt2  44185  caucvgbf  44498  limsuppnfd  44716  limsuppnf  44725  limsupre2  44739  limsupre3  44747  limsupre3uz  44750  limsupreuz  44751  liminfreuz  44817  stoweidlem31  45045  stoweidlem59  45073  rexsb  46105  cbvrex2  46110  2reu8i  46119