Theorem pm13.194 40295

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 40290	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2		sbsbc 3711	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylibr 235	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4		simpl 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝜑)
5		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
6	3, 4, 5	3jca 1121	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))
7		3simpc 1143	. 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦) → (𝜑 ∧ 𝑥 = 𝑦))
8	6, 7	impbii 210	1 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1080  [wsb 2041  [wsbc 3707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-12 2140  ax-ext 2768

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1082  df-ex 1763  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-sbc 3708

This theorem is referenced by: (None)