Theorem ch2varv 10579

Description: Version of ch2var 10578 with non-freeness hypotheses replaced by DV 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 1461	. 2 ⊢ Ⅎ𝑥𝜓
2		nfv 1461	. 2 ⊢ Ⅎ𝑧𝜓
3		ch2varv.maj	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))
4		ch2varv.min	. 2 ⊢ 𝜑
5	1, 2, 3, 4	ch2var 10578	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  sscoll2  10783