Theorem ch2var 9907

Description: Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

Ref	Expression
ch2var.nfx	⊢ Ⅎ𝑥𝜓
ch2var.nfz	⊢ Ⅎ𝑧𝜓
ch2var.maj	⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))
ch2var.min	⊢ 𝜑

Assertion

Ref	Expression
ch2var	⊢ 𝜓

Distinct variable groups:   𝑥,𝑧   𝑥,𝑡

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

Proof of Theorem ch2var

Step	Hyp	Ref	Expression
1		ch2var.nfx	. . 3 ⊢ Ⅎ𝑥𝜓
2		ch2var.nfz	. . 3 ⊢ Ⅎ𝑧𝜓
3		ch2var.maj	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))
4	3	biimpd 132	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 → 𝜓))
5	1, 2, 4	2spim 9906	. 2 ⊢ (∀𝑧∀𝑥𝜑 → 𝜓)
6		ch2var.min	. . 3 ⊢ 𝜑
7	6	ax-gen 1338	. 2 ⊢ ∀𝑥𝜑
8	5, 7	mpg 1340	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  ch2varv  9908