Theorem 2pm13.193 44544

Description: pm13.193 44402 for two variables. pm13.193 44402 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 44894. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
2pm13.193	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Proof of Theorem 2pm13.193

Step	Hyp	Ref	Expression
1		simpll 766	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝑥 = 𝑢)
2		simplr 768	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝑦 = 𝑣)
3		simpr 484	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4		sbequ2 2250	. . . . 5 ⊢ (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
5	1, 3, 4	sylc 65	. . . 4 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑣 / 𝑦]𝜑)
6		sbequ2 2250	. . . 4 ⊢ (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑 → 𝜑))
7	2, 5, 6	sylc 65	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → 𝜑)
8	1, 2, 7	jca31 514	. 2 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
9		simpll 766	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → 𝑥 = 𝑢)
10		simplr 768	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → 𝑦 = 𝑣)
11		simpr 484	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
12		sbequ1 2249	. . . . 5 ⊢ (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
13	10, 11, 12	sylc 65	. . . 4 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → [𝑣 / 𝑦]𝜑)
14		sbequ1 2249	. . . 4 ⊢ (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
15	9, 13, 14	sylc 65	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
16	9, 10, 15	jca31 514	. 2 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
17	8, 16	impbii 209	1 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  [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  ax-12 2178

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

This theorem is referenced by:  2sb5nd  44552  2sb5ndVD  44901  2sb5ndALT  44923