Theorem cbvaldva 2413

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvaldvaw 2036 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvaldva.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvaldva	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvaldva

Step	Hyp	Ref	Expression
1		nfv 1913	. 2 ⊢ Ⅎ𝑦𝜑
2		nfvd 1914	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3		cbvaldva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	3	ex 412	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5	1, 2, 4	cbvald 2411	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-11 2156  ax-12 2176  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783

This theorem is referenced by:  cbval2vv  2417