Theorem sbequ 2084

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 2066. (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 2026	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢 = 𝑥 ↔ 𝑢 = 𝑦))
2	1	imbi1d 341	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)) ↔ (𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑))))
3	2	albidv 1920	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑢(𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)) ↔ ∀𝑢(𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑))))
4		df-sb 2066	. 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ ∀𝑢(𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)))
5		df-sb 2066	. 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∀𝑢(𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑)))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066

This theorem is referenced by:  sbequi  2085  sbcom3vv  2098  sbco2vv  2100  sbco4lem  2102  sbco4  2103  sbcom2  2174  drsb2  2267  sbco2v  2330  sbcom3  2504  sbco2  2509  sb10f  2525  sb8eulem  2591  eleq1ab  2709  cbvralsvwOLDOLD  3290  cbvrexsvwOLD  3291  cbvralf  3331  cbvralsv  3337  cbvrexsv  3338  cbvreu  3394  cbvrabwOLD  3439  cbvrab  3443  cbvreucsf  3903  cbvrabcsf  3904  cbvopab1g  5177  cbvmptf  5202  cbvmptfg  5203  cbviota  6461  sb8iota  6463  cbvriota  7339  tfis  7811  tfinds  7816  findes  7856  uzind4s  12843  wl-sbcom2d-lem1  37520  wl-sb8eut  37539  wl-sb8eutv  37540  wl-dfclab  37557  sbeqi  38126  disjinfi  45159  2reu8i  47087