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  3293  cbvrexsvwOLD  3294  cbvralf  3334  cbvralsv  3340  cbvrexsv  3341  cbvreu  3397  cbvrabwOLD  3442  cbvrab  3446  cbvreucsf  3906  cbvrabcsf  3907  cbvopab1g  5182  cbvmptf  5207  cbvmptfg  5208  cbviota  6473  sb8iota  6475  cbvriota  7357  tfis  7831  tfinds  7836  findes  7876  uzind4s  12867  wl-sbcom2d-lem1  37547  wl-sb8eut  37566  wl-sb8eutv  37567  wl-dfclab  37584  sbeqi  38153  disjinfi  45186  2reu8i  47114