Theorem cbvalv1 2339

Description: Rule used to change bound variables, using implicit substitution. Version of cbval 2396 with a disjoint variable condition, which does not require ax-13 2370. See cbvalvw 2036 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2398 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 2333	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	biimprd 248	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7	6	equcoms 2020	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8	2, 1, 7	cbv3v 2333	. 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 2008  ax-11 2158  ax-12 2178

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

This theorem is referenced by:  cbvexv1  2340  cbval2v  2341  sbbib  2359  cbvsbvf  2361  sb8eulem  2591  cbvmow  2596  abbib  2798  cleqh  2857  cleqf  2920  cbvralfw  3270  cbvralf  3325  ralab2  3659  cbvralcsf  3895  dfssf  3928  elintabOLD  4912  reusv2lem4  5343  cbviotaw  6449  cbviota  6451  sb8iota  6453  dffun6f  6501  findcard2  9088  aceq1  10030  bnj1385  34798  sbcalf  38093  alrimii  38098  aomclem6  43032  rababg  43547