Theorem pm13.194 45150

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 45145	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2		sbsbc 3748	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 237	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4		simpl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
5		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
6	3, 4, 5	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))
7		3simpc 1168	. 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦) → (𝜑 ∧ 𝑥 = 𝑦))
8	6, 7	impbii 212	1 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  [wsb 2096  [wsbc 3744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-sbc 3745

This theorem is used by: (None)