Theorem sbequ 2088

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 2070. (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 2033	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢 = 𝑥 ↔ 𝑢 = 𝑦))
2	1	imbi1d 345	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)) ↔ (𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑))))
3	2	albidv 1921	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑢(𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)) ↔ ∀𝑢(𝑢 = 𝑦 → ∀𝑧(𝑧 = 𝑢 → 𝜑))))
4		df-sb 2070	. 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ ∀𝑢(𝑢 = 𝑥 → ∀𝑧(𝑧 = 𝑢 → 𝜑)))
5		df-sb 2070	. 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))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070

This theorem is referenced by:  sbequi  2089  sbcom3vv  2103  sbco2vv  2105  sbcom2  2165  drsb2  2264  sbco2v  2341  sbcom3  2525  sbco2  2530  sb10f  2547  sb8eulem  2659  eleq1ab  2778  cbvralfwOLD  3383  cbvralf  3385  cbvreuw  3389  cbvreu  3394  cbvralsvw  3414  cbvrexsvw  3415  cbvralsv  3416  cbvrexsv  3417  cbvrabw  3437  cbvrab  3438  cbvreucsf  3872  cbvrabcsf  3873  ss2abdv  3991  cbvopab1g  5104  cbvmptf  5129  cbvmptfg  5130  cbviota  6292  sb8iota  6294  cbvriota  7106  tfis  7549  tfinds  7554  findes  7593  uzind4s  12296  wl-sbcom2d-lem1  34960  wl-sb8eut  34978  wl-dfclab  34993  sbeqi  35597  disjinfi  41820  2reu8i  43669