Theorem drsb1 1809

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
drsb1	⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))

Proof of Theorem drsb1

Step	Hyp	Ref	Expression
1		equequ1 1722	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
2	1	sps 1547	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
3	2	imbi1d 231	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → 𝜑)))
4	2	anbi1d 465	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧 ∧ 𝜑) ↔ (𝑦 = 𝑧 ∧ 𝜑)))
5	4	drex1 1808	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)))
6	3, 5	anbi12d 473	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ ((𝑦 = 𝑧 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑))))
7		df-sb 1773	. 2 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))
8		df-sb 1773	. 2 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ((𝑦 = 𝑧 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)))
9	6, 7, 8	3bitr4g 223	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1361  ∃wex 1502  [wsb 1772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-sb 1773

This theorem is referenced by:  sbequi  1849  nfsbxy  1953  nfsbxyt  1954  sbcomxyyz  1983  iotaeq  5200