Theorem alcomiw 1958

Description: Weak version of alcom 2024. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
alcomiw.1	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
alcomiw	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Distinct variable groups:   𝑦,𝑧   𝑥,𝑦   𝜑,𝑧   𝜓,𝑦

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

Proof of Theorem alcomiw

Step	Hyp	Ref	Expression
1		alcomiw.1	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
2	1	biimpd 218	. . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 → 𝜓))
3	2	cbvalivw 1921	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑧𝜓)
4	3	alimi 1730	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑧𝜓)
5		ax-5 1827	. 2 ⊢ (∀𝑥∀𝑧𝜓 → ∀𝑦∀𝑥∀𝑧𝜓)
6	1	biimprd 237	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜓 → 𝜑))
7	6	equcoms 1934	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝜓 → 𝜑))
8	7	spimvw 1914	. . . 4 ⊢ (∀𝑧𝜓 → 𝜑)
9	8	alimi 1730	. . 3 ⊢ (∀𝑥∀𝑧𝜓 → ∀𝑥𝜑)
10	9	alimi 1730	. 2 ⊢ (∀𝑦∀𝑥∀𝑧𝜓 → ∀𝑦∀𝑥𝜑)
11	4, 5, 10	3syl 18	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  hbalw  1964  ax11w  1994  bj-ssblem2  31820