Theorem ch2varv 16796

Description: Version of ch2var 16795 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

Ref	Expression
ch2varv.maj	⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))
ch2varv.min	⊢ 𝜑

Assertion

Ref	Expression
ch2varv	⊢ 𝜓

Distinct variable groups:   𝑥,𝑧,𝜓   𝑥,𝑡

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

Proof of Theorem ch2varv

Step	Hyp	Ref	Expression
1		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜓
2		nfv 1581	. 2 ⊢ Ⅎ𝑧𝜓
3		ch2varv.maj	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))
4		ch2varv.min	. 2 ⊢ 𝜑
5	1, 2, 3, 4	ch2var 16795	1 ⊢ 𝜓

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  sscoll2  17014