Theorem pm13.194 45221

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 45216	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2		sbsbc 3746	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 237	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4		simpl 488	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
5		simpr 490	. . 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 401   ∧ w3a 1103  [wsb 2099  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3743

This theorem is used by: (None)