Theorem cbval 1803

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbval	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1568	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbval.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1568	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbval.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvalh 1802	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396  Ⅎwnf 1509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510

This theorem is referenced by:  sb8  1905  cbval2  1973  sb8eu  2095  abbibcom  2348  cleqf  2411  cbvralf  2771  ralab2  2983  cbvralcsf  3203  dfss2f  3231  elintab  3962  cbviota  5319  sb8iota  5322  dffun6f  5367  dffun4f  5370  mptfvex  5765  findcard2  7148  findcard2s  7149