Theorem sbequ 2120

Description: Equality property for substitution, from Tarski's system. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) Revise df-sb 2100. (Revised by BJ, 30-Dec-2020.)

Assertion

Ref	Expression
sbequ	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequ

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 2059	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢 = 𝑥 ↔ 𝑢 = 𝑦))
2	1	imbi1d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)) ↔ (𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑))))
3	2	albidv 1953	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑢(𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)) ↔ ∀𝑢(𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑))))
4		dfsb 2101	. 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ ∀𝑢(𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)))
5		dfsb 2101	. 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∀𝑢(𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑)))
6	3, 4, 5	3bitr4g 317	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  sbequiOLD  2121  sbcom3vv  2134  sbco2vv  2136  sbco4lem  2138  sbco4  2139  sbcom2  2209  drsb2  2301  sbco2v  2362  sbcom3  2536  sbco2  2541  sb10f  2557  sb8eulem  2624  eleq1ab  2741  cbvralf  3346  cbvralsv  3352  cbvrexsv  3353  cbvreu  3405  cbvrab  3450  cbvreucsf  3891  cbvrabcsf  3892  cbvopab1g  5180  cbvmptfg  5206  cbviota  6502  sb8iota  6504  cbvriota  7388  tfis  7864  tfinds  7869  findes  7910  uzind4s  13028  regsfromregtco  37306  bj-axseprep  37970  wl-sbcom2d-lem1  38471  wl-sb8eut  38490  wl-sb8eutv  38491  wl-dfclab  38497  sbeqi  39071  disjinfi  46176  2reu8i  48152