Theorem cbvalv1 2342

Description: Rule used to change bound variables, using implicit substitution. Version of cbval 2402 with a disjoint variable condition, which does not require ax-13 2376. See cbvalvw 2035 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2404 for another variant. (Contributed by NM, 13-May-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
cbvalv1	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvalv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . 3 ⊢ Ⅎ𝑦𝜑
2		cbvalv1.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3		cbvalv1.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	biimpd 229	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbv3v 2336	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	biimprd 248	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7	6	equcoms 2019	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8	2, 1, 7	cbv3v 2336	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
9	5, 8	impbii 209	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  Ⅎwnf 1783

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 1910  ax-6 1967  ax-7 2007  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbvexv1  2343  cbval2v  2344  sb8fOLD  2356  sbbib  2363  cbvsbvf  2365  sb8eulem  2597  cbvmow  2602  abbib  2804  cleqh  2864  cleqf  2927  cbvralfw  3284  cbvralf  3339  ralab2  3680  cbvralcsf  3916  dfssf  3949  elintabOLD  4935  reusv2lem4  5371  cbviotaw  6490  cbviota  6492  sb8iota  6494  dffun6f  6548  findcard2  9176  aceq1  10129  bnj1385  34809  sbcalf  38084  alrimii  38089  aomclem6  43030  rababg  43545