Theorem cbvexv1 2371

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex 2428 with a disjoint variable condition, which does not require ax-13 2401. See cbvexvw 2070 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2430 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	⊢ Ⅎ𝑦𝜑
cbvalv1.nf2	⊢ Ⅎ𝑥𝜓
cbvalv1.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvexv1	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvexv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nfn 1890	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbvalv1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1890	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbvalv1.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 321	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbvalv1 2370	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
9		alnex 1814	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
10	7, 8, 9	3bitr3i 304	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓)
11	10	con4bii 324	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  sb8ef  2384  exsb  2388  mof  2588  euf  2601  cbveuw  2631  eqvincf  3603  rexab2  3656  euabsn  4686  eluniab  4880  cbvopab1  5178  cbvopab1g  5179  cbvopab2  5180  cbvopab1s  5181  axrep1  5232  axrep2  5234  opeliunxp  5714  opeliun2xp  5715  dfdmf  5874  dfrnf  5928  elrnmpt1  5938  cbvoprab1  7495  cbvoprab2  7496  opabex3d  7960  opabex3rd  7961  opabex3  7962  zfcndrep  10671  fsum2dlem  15904  fprod2dlem  16115  2ndresdju  33177  bnj1146  35356  bnj607  35481  bnj1228  35576  fineqvrep  35707  poimirlem26  38484  sbcexf  38967  elunif  45954  stoweidlem46  46978