Theorem 2spim 10577

Description: Double substitution, as in spim 1666. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

Ref	Expression
2spim.nfx	⊢ Ⅎ𝑥𝜒
2spim.nfz	⊢ Ⅎ𝑧𝜒
2spim.1	⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))

Assertion

Ref	Expression
2spim	⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

Distinct variable groups:   𝑥,𝑧   𝑥,𝑡

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

Proof of Theorem 2spim

Step	Hyp	Ref	Expression
1		2spim.nfz	. 2 ⊢ Ⅎ𝑧𝜒
2		2spim.nfx	. . . 4 ⊢ Ⅎ𝑥𝜒
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑡 → Ⅎ𝑥𝜒)
4		2spim.1	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))
5	4	expcom 114	. . . 4 ⊢ (𝑧 = 𝑡 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	5	alrimiv 1795	. . 3 ⊢ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))
7	3, 6	spimd 10576	. 2 ⊢ (𝑧 = 𝑡 → (∀𝑥𝜓 → 𝜒))
8	1, 7	spim 1666	1 ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282  Ⅎwnf 1389

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:  ch2var  10578