Theorem 2spim 16806

Description: Double substitution, as in spim 1791. (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 116	. . . 4 ⊢ (𝑧 = 𝑡 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	5	alrimiv 1927	. . 3 ⊢ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))
7	3, 6	spimd 16805	. 2 ⊢ (𝑧 = 𝑡 → (∀𝑥𝜓 → 𝜒))
8	1, 7	spim 1791	1 ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wal 1400  Ⅎwnf 1513

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:  ch2var  16807