Theorem pm13.194 38613

Description: Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.194	⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))

Proof of Theorem pm13.194

Step	Hyp	Ref	Expression
1		pm13.13a 38608	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2		sbsbc 3439	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 224	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4		simpl 473	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
5		simpr 477	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
6	3, 4, 5	3jca 1242	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))
7		3simpc 1060	. 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦) → (𝜑 ∧ 𝑥 = 𝑦))
8	6, 7	impbii 199	1 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  [wsb 1880  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by: (None)